gdz.top
Номер 853, страница 218 - гдз по геометрии 10-11 класс учебник Атанасян, Бутузов
Авторы: Атанасян Л. С., Бутузов В. Ф., Кадомцев С. Б., Позняк Э. Г., Киселёва Л. С.
Тип: Учебник
Издательство: Просвещение
Год издания: 2019 - 2026
Уровень обучения: базовый и углублённый
Цвет обложки: коричневый с ромбами
ISBN: 978-5-09-103606-0 (2023)
Допущено Министерством просвещения Российской Федерации
Математика: алгебра и начала математического анализа, геометрия
Популярные ГДЗ в 10 классе
Глава 8. Некоторые сведения из планиметрии. Параграф 3. Теоремы Менелая и Чевы - номер 853, страница 218.
App\Models\Task {#1030 // resources/views/models/task/default.blade.php #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 745268 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "218" "field_page_end" => null "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/00-853" "field_display_title" => "853" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037 #items: array:1 [ 0 => App\Models\Term {#1036 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #original: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1035 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1046 #items: array:1 [ 0 => App\Models\Branch {#1045 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 744411 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Некоторые сведения из планиметрии" "field_branch_order" => "8" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1050 #items: array:1 [ 0 => App\Models\Term {#1047 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => null ] #original: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "194" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1086 #items: array:1 [ 0 => App\Models\Book {#1049 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 36 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1051 #items: array:1 [ 0 => App\Models\Term {#1048 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6477 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Геометрия" "field_abbreviated_name" => null "field_cases" => array:6 [ …6] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "geometrija" ] #original: array:10 [ "id" => 6477 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Геометрия" "field_abbreviated_name" => null "field_cases" => array:6 [ …6] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "geometrija" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1052 #items: array:2 [ 0 => App\Models\Term {#1053 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ …6] "field_translit" => "desjatyj" ] #original: array:6 [ "id" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ …6] "field_translit" => "desjatyj" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Term {#1054 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5460 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "11" "field_cases" => array:6 [ …6] "field_translit" => "odinadcatyj" ] #original: array:6 [ "id" => 5460 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "11" "field_cases" => array:6 [ …6] "field_translit" => "odinadcatyj" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1055 #items: array:1 [ 0 => App\Models\Term {#1056 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5153 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Просвещение" "field_cases" => null "field_translit" => "prosveschenie" ] #original: array:6 [ "id" => 5153 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Просвещение" "field_cases" => null "field_translit" => "prosveschenie" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1057 #items: array:5 [ 0 => App\Models\Term {#1058 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 3753 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Атанасян" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Левон" "field_patronymic" => "Сергеевич" "field_surname_rp" => null ] #original: array:12 [ "id" => 3753 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Атанасян" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Левон" "field_patronymic" => "Сергеевич" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Term {#1059 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 3949 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Бутузов" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Валентин" "field_patronymic" => "Фёдорович" "field_surname_rp" => null ] #original: array:12 [ "id" => 3949 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Бутузов" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Валентин" "field_patronymic" => "Фёдорович" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Term {#1060 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 4518 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кадомцев" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Сергей" "field_patronymic" => "Борисович" "field_surname_rp" => null ] #original: array:12 [ "id" => 4518 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кадомцев" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Сергей" "field_patronymic" => "Борисович" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 3 => App\Models\Term {#1061 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 5494 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Позняк" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Эдуард" "field_patronymic" => "Генрихович" "field_surname_rp" => null ] #original: array:12 [ "id" => 5494 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Позняк" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Эдуард" "field_patronymic" => "Генрихович" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 4 => App\Models\Term {#1062 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 6905 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Киселёва" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Людмила" "field_patronymic" => "Сергеевна" "field_surname_rp" => null ] #original: array:12 [ "id" => 6905 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Киселёва" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Людмила" "field_patronymic" => "Сергеевна" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1063 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1064 #items: array:1 [ 0 => App\Models\Term {#1065 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ …6] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #original: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ …6] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1066 #items: array:1 [ 0 => App\Models\Term {#1067 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 9 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Россия" "field_cases" => array:6 [ …6] "field_translit" => "rossiya" ] #original: array:6 [ "id" => 9 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Россия" "field_cases" => array:6 [ …6] "field_translit" => "rossiya" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1068 #items: array:1 [ 0 => App\Models\Term {#1069 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 15 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Москва" "field_cases" => array:6 [ …6] "field_translit" => "moskva" ] #original: array:6 [ "id" => 15 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Москва" "field_cases" => array:6 [ …6] "field_translit" => "moskva" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1070 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1071 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1072 #items: array:1 [ 0 => App\Models\Term {#1073 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 41 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "базовый и углублённый" "field_cases" => array:6 [ …6] "field_translit" => "bazovyy i uglublennyy" ] #original: array:6 [ "id" => 41 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "базовый и углублённый" "field_cases" => array:6 [ …6] "field_translit" => "bazovyy i uglublennyy" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1074 #items: array:1 [ 0 => App\Models\Term {#1075 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6706 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "ФГОС (старый)" "field_cases" => array:6 [ …6] "field_color" => null "field_end_year" => "2022" "field_start_year" => "2019" "field_translit" => "fgos-old" "field_type" => null ] #original: array:10 [ "id" => 6706 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "ФГОС (старый)" "field_cases" => array:6 [ …6] "field_color" => null "field_end_year" => "2022" "field_start_year" => "2019" "field_translit" => "fgos-old" "field_type" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_publication_number" => "11" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1076 #items: array:1 [ 0 => App\Models\Term {#1077 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 33 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "стереотипное" "field_cases" => array:6 [ …6] "field_translit" => "stereotipnoe" ] #original: array:6 [ "id" => 33 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "стереотипное" "field_cases" => array:6 [ …6] "field_translit" => "stereotipnoe" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1078 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => "с ромбами" "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для общеобразовательных организаций" "field_allowed" => "Допущено Министерством просвещения Российской Федерации" "field_reserve_field" => "Математика: алгебра и начала математического анализа, геометрия" "field_link_to_source" => null "field_tasks_count" => "1147" "field_priority" => "12" "field_default_folder" => "/geometrija_10/atanasjan-u23/" "field_isbn" => "978-5-09-103606-0 (2023)" "field_cover" => array:1 [ 0 => "/media/geometrija_10/atanasjan-u23/covers/cover1.webp?ts=1739363242" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/geometrija_10/atanasjan-u23/covers/cover1.webp?ts=1739363242" "alt" => "" "width" => "1200" "height" => "1661" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1079 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1080 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1081 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/geometrija/atanasjan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1082 #items: array:1 [ 0 => App\Models\Term {#1083 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 51 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "коричневый" "field_cases" => array:6 [ …6] "field_translit" => "korichnevyy" ] #original: array:6 [ "id" => 51 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "коричневый" "field_cases" => array:6 [ …6] "field_translit" => "korichnevyy" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 543 "class_subject" => 392 ] ] #original: array:50 [ "id" => 36 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1051} "field_class" => Illuminate\Database\Eloquent\Collection {#1052} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1055} "field_author" => Illuminate\Database\Eloquent\Collection {#1057} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1063} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1064} "field_country" => Illuminate\Database\Eloquent\Collection {#1066} "field_city" => Illuminate\Database\Eloquent\Collection {#1068} "field_series" => Illuminate\Database\Eloquent\Collection {#1070} "field_umk" => Illuminate\Database\Eloquent\Collection {#1071} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1072} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1074} "field_publication_number" => "11" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1076} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1078} "field_under_the_edition_degree" => null "field_cover_description" => "с ромбами" "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для общеобразовательных организаций" "field_allowed" => "Допущено Министерством просвещения Российской Федерации" "field_reserve_field" => "Математика: алгебра и начала математического анализа, геометрия" "field_link_to_source" => null "field_tasks_count" => "1147" "field_priority" => "12" "field_default_folder" => "/geometrija_10/atanasjan-u23/" "field_isbn" => "978-5-09-103606-0 (2023)" "field_cover" => array:1 [ 0 => "/media/geometrija_10/atanasjan-u23/covers/cover1.webp?ts=1739363242" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/geometrija_10/atanasjan-u23/covers/cover1.webp?ts=1739363242" "alt" => "" "width" => "1200" "height" => "1661" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1079} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1080} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1081} "field_url" => "/10-klass/geometrija/atanasjan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1082} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 543 "class_subject" => 392 ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1084 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 744411 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Некоторые сведения из планиметрии" "field_branch_order" => "8" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1050} "field_page_start" => "194" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1086} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1084} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1121 #items: array:2 [ 0 => App\Models\Branch {#1129 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 744411 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Некоторые сведения из планиметрии" "field_branch_order" => "8" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1130 #items: array:1 [ 0 => App\Models\Term {#1047} ] #escapeWhenCastingToString: false } "field_page_start" => "194" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1131 #items: array:1 [ 0 => App\Models\Book {#1049} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1132 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 744411 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Некоторые сведения из планиметрии" "field_branch_order" => "8" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1130} "field_page_start" => "194" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1131} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1132} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Branch {#1133 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 744414 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Теоремы Менелая и Чевы" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1137 #items: array:1 [ 0 => App\Models\Term {#1134 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => "§" ] #original: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => "§" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "214" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1136 #items: array:1 [ 0 => App\Models\Book {#1049} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1135 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 744414 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Теоремы Менелая и Чевы" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1137} "field_page_start" => "214" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1136} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1135} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1038 #items: array:4 [ 0 => App\Models\Element {#1111 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 940227 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1100 #items: array:1 [ 0 => App\Models\Edition {#1108 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 2315 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1106 …2} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1107 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1105 …2} "field_process_formula" => "wiris" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1104 …2} "field_content_source" => null ] #original: array:21 [ "id" => 2315 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1106 …2} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1107 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1105 …2} "field_process_formula" => "wiris" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1104 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "745268" "type" => "task" ] "text" => "<p> <b>853.</b> На сторонах ВС, СА и АВ треугольника АВС или их продолжениях отмечены соответственно точки А₁, В₁ и С₁, лежащие на одной прямой. Докажите, что точки А₂, В₂ и С₂, симметричные соответственно точкам А₁, В₁ и С₁ относительно середин сторон ВС, СА и АВ, также лежат на одной прямой. </p>" "img" => array:1 [ 0 => array:5 [ "name" => "853-1.jpg" "alt" => null "width" => "1500" "height" => 293 "path" => "/media/geometrija_10/atanasjan-u23/0-00/853-1.webp?ts=1739436954" ] ] ] #original: array:7 [ "id" => 940227 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1100} "task" => array:2 [ "refs" => "745268" "type" => "task" ] "text" => "<p> <b>853.</b> На сторонах ВС, СА и АВ треугольника АВС или их продолжениях отмечены соответственно точки А₁, В₁ и С₁, лежащие на одной прямой. Докажите, что точки А₂, В₂ и С₂, симметричные соответственно точкам А₁, В₁ и С₁ относительно середин сторон ВС, СА и АВ, также лежат на одной прямой. </p>" "img" => array:1 [ 0 => array:5 [ "name" => "853-1.jpg" "alt" => null "width" => "1500" "height" => 293 "path" => "/media/geometrija_10/atanasjan-u23/0-00/853-1.webp?ts=1739436954" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Element {#1101 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 942814 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1088 #items: array:1 [ 0 => App\Models\Edition {#1098 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 2317 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1099 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1096 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1097 …2} "field_content_source" => null ] #original: array:21 [ "id" => 2317 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1099 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1096 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1097 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "745268" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "853-1.png" "alt" => null "width" => "694" "height" => 2631 "path" => "/media/geometrija_10/atanasjan-u23/2-00/853-1.webp?ts=1739441758" ] ] ] #original: array:6 [ "id" => 942814 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1088} "task" => array:2 [ "refs" => "745268" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "853-1.png" "alt" => null "width" => "694" "height" => 2631 "path" => "/media/geometrija_10/atanasjan-u23/2-00/853-1.webp?ts=1739441758" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Element {#1102 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 942547 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1093 #items: array:1 [ 0 => App\Models\Edition {#1094 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 2318 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 3" "field_order" => "4" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1039 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "3-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1043 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1090 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1092 …2} "field_content_source" => null ] #original: array:21 [ "id" => 2318 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 3" "field_order" => "4" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1039 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "3-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1043 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1090 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1092 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "745268" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "853-1.jpg" "alt" => null "width" => "1399" "height" => 849 "path" => "/media/geometrija_10/atanasjan-u23/3-00/853-1.webp?ts=1739437243" ] ] ] #original: array:6 [ "id" => 942547 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1093} "task" => array:2 [ "refs" => "745268" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "853-1.jpg" "alt" => null "width" => "1399" "height" => 849 "path" => "/media/geometrija_10/atanasjan-u23/3-00/853-1.webp?ts=1739437243" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 3 => App\Models\Element {#1089 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1352581 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1138 #items: array:1 [ 0 => App\Models\Edition {#1087 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 5120 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 6" "field_order" => "7" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1120 …2} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "vadim" "field_edition_type" => "solution" "field_root_dir" => "6-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1119 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1124 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1122 …2} "field_content_source" => null ] #original: array:21 [ "id" => 5120 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 6" "field_order" => "7" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1120 …2} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "vadim" "field_edition_type" => "solution" "field_root_dir" => "6-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1119 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1124 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1122 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "745268" "type" => "task" ] "text" => "<p>Для решения задачи воспользуемся векторным методом в сочетании с теоремой Менелая.</p><p>Пусть $O$ — произвольная точка плоскости (начало координат). Обозначим радиус-векторы вершин треугольника $A, B, C$ как $\vec{a}, \vec{b}, \vec{c}$ соответственно. Пусть $M_a, M_b, M_c$ — середины сторон $BC, CA, AB$. Их радиус-векторы равны:</p><p>$\vec{m}_a = \frac{\vec{b}+\vec{c}}{2}$, $\vec{m}_b = \frac{\vec{c}+\vec{a}}{2}$, $\vec{m}_c = \frac{\vec{a}+\vec{b}}{2}$</p><p>Точка $A_1$ лежит на прямой $BC$, поэтому ее радиус-вектор $\vec{a_1}$ можно представить как аффинную комбинацию векторов $\vec{b}$ и $\vec{c}$: $\vec{a_1} = (1-t_a)\vec{b} + t_a\vec{c}$ для некоторого скалярного параметра $t_a$. Аналогично для точек $B_1$ на прямой $CA$ и $C_1$ на прямой $AB$:</p><p>$\vec{b_1} = (1-t_b)\vec{c} + t_b\vec{a}$</p><p>$\vec{c_1} = (1-t_c)\vec{a} + t_c\vec{b}$</p><p>где $t_b$ и $t_c$ — также скалярные параметры.</p><p>Точка $A_2$ симметрична точке $A_1$ относительно середины $M_a$ стороны $BC$. Это означает, что $M_a$ является серединой отрезка $A_1A_2$. В векторной форме это условие записывается как $\vec{m}_a = \frac{\vec{a_1}+\vec{a_2}}{2}$. Отсюда можно выразить радиус-вектор точки $A_2$:</p><p>$\vec{a_2} = 2\vec{m}_a - \vec{a_1} = 2 \left( \frac{\vec{b}+\vec{c}}{2} \right) - \vec{a_1} = \vec{b}+\vec{c} - \vec{a_1}$</p><p>Подставим в это равенство выражение для $\vec{a_1}$:</p><p>$\vec{a_2} = \vec{b}+\vec{c} - ((1-t_a)\vec{b} + t_a\vec{c}) = (1 - (1-t_a))\vec{b} + (1-t_a)\vec{c} = t_a\vec{b} + (1-t_a)\vec{c}$</p><p>Аналогично найдем радиус-векторы для точек $B_2$ и $C_2$:</p><p>$\vec{b_2} = 2\vec{m}_b - \vec{b_1} = \vec{c}+\vec{a} - \vec{b_1} = \vec{c}+\vec{a} - ((1-t_b)\vec{c} + t_b\vec{a}) = t_b\vec{c} + (1-t_b)\vec{a}$</p><p>$\vec{c_2} = 2\vec{m}_c - \vec{c_1} = \vec{a}+\vec{b} - \vec{c_1} = \vec{a}+\vec{b} - ((1-t_c)\vec{a} + t_c\vec{b}) = t_c\vec{a} + (1-t_c)\vec{b}$</p><p>По условию задачи, точки $A_1, B_1, C_1$ лежат на одной прямой. Согласно теореме Менелая, это эквивалентно выполнению следующего равенства для отношений, в которых эти точки делят стороны треугольника (с учетом знака, то есть для ориентированных отрезков):</p><p>$\frac{\vec{BA_1}}{\vec{A_1C}} \cdot \frac{\vec{CB_1}}{\vec{B_1A}} \cdot \frac{\vec{AC_1}}{\vec{C_1B}} = 1$</p><p>Выразим эти отношения через параметры $t_a, t_b, t_c$.</p><p>Для точки $A_1$: $\vec{BA_1} = \vec{a_1} - \vec{b} = t_a(\vec{c}-\vec{b}) = t_a\vec{BC}$ и $\vec{A_1C} = \vec{c} - \vec{a_1} = (1-t_a)(\vec{c}-\vec{b}) = (1-t_a)\vec{BC}$. Отсюда $\frac{\vec{BA_1}}{\vec{A_1C}} = \frac{t_a}{1-t_a}$.</p><p>Аналогично, для точки $B_1$: $\frac{\vec{CB_1}}{\vec{B_1A}} = \frac{t_b}{1-t_b}$.</p><p>И для точки $C_1$: $\frac{\vec{AC_1}}{\vec{C_1B}} = \frac{t_c}{1-t_c}$.</p><p>Таким образом, условие коллинеарности точек $A_1, B_1, C_1$ принимает вид:</p><p>$\frac{t_a}{1-t_a} \cdot \frac{t_b}{1-t_b} \cdot \frac{t_c}{1-t_c} = 1$</p><p>Теперь проверим, выполняется ли условие теоремы Менелая для точек $A_2, B_2, C_2$. Для этого вычислим соответствующие отношения отрезков.</p><p>Для точки $A_2$: $\vec{BA_2} = \vec{a_2}-\vec{b} = (t_a-1)\vec{b} + (1-t_a)\vec{c} = (1-t_a)(\vec{c}-\vec{b}) = (1-t_a)\vec{BC}$ и $\vec{A_2C} = \vec{c}-\vec{a_2} = -t_a\vec{b} + t_a\vec{c} = t_a(\vec{c}-\vec{b}) = t_a\vec{BC}$. Отсюда $\frac{\vec{BA_2}}{\vec{A_2C}} = \frac{1-t_a}{t_a}$.</p><p>Аналогично, для точки $B_2$: $\frac{\vec{CB_2}}{\vec{B_2A}} = \frac{1-t_b}{t_b}$.</p><p>И для точки $C_2$: $\frac{\vec{AC_2}}{\vec{C_2B}} = \frac{1-t_c}{t_c}$.</p><p>Теперь найдем произведение этих отношений:</p><p>$\frac{\vec{BA_2}}{\vec{A_2C}} \cdot \frac{\vec{CB_2}}{\vec{B_2A}} \cdot \frac{\vec{AC_2}}{\vec{C_2B}} = \frac{1-t_a}{t_a} \cdot \frac{1-t_b}{t_b} \cdot \frac{1-t_c}{t_c}$</p><p>Это произведение является обратным к произведению для точек $A_1, B_1, C_1$:</p><p>$\frac{1-t_a}{t_a} \cdot \frac{1-t_b}{t_b} \cdot \frac{1-t_c}{t_c} = \left( \frac{t_a}{1-t_a} \cdot \frac{t_b}{1-t_b} \cdot \frac{t_c}{1-t_c} \right)^{-1}$</p><p>Поскольку мы установили, что произведение для коллинеарных точек $A_1, B_1, C_1$ равно 1, то и для $A_2, B_2, C_2$ получаем:</p><p>$\left(1\right)^{-1} = 1$</p><p>Так как для точек $A_2, B_2, C_2$ выполняется условие теоремы Менелая, то по обратной теореме Менелая эти точки лежат на одной прямой. Что и требовалось доказать.</p><p>Ответ: Точки $A_2, B_2, C_2$ также лежат на одной прямой.</p>" ] #original: array:6 [ "id" => 1352581 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1138} "task" => array:2 [ "refs" => "745268" "type" => "task" ] "text" => "<p>Для решения задачи воспользуемся векторным методом в сочетании с теоремой Менелая.</p><p>Пусть $O$ — произвольная точка плоскости (начало координат). Обозначим радиус-векторы вершин треугольника $A, B, C$ как $\vec{a}, \vec{b}, \vec{c}$ соответственно. Пусть $M_a, M_b, M_c$ — середины сторон $BC, CA, AB$. Их радиус-векторы равны:</p><p>$\vec{m}_a = \frac{\vec{b}+\vec{c}}{2}$, $\vec{m}_b = \frac{\vec{c}+\vec{a}}{2}$, $\vec{m}_c = \frac{\vec{a}+\vec{b}}{2}$</p><p>Точка $A_1$ лежит на прямой $BC$, поэтому ее радиус-вектор $\vec{a_1}$ можно представить как аффинную комбинацию векторов $\vec{b}$ и $\vec{c}$: $\vec{a_1} = (1-t_a)\vec{b} + t_a\vec{c}$ для некоторого скалярного параметра $t_a$. Аналогично для точек $B_1$ на прямой $CA$ и $C_1$ на прямой $AB$:</p><p>$\vec{b_1} = (1-t_b)\vec{c} + t_b\vec{a}$</p><p>$\vec{c_1} = (1-t_c)\vec{a} + t_c\vec{b}$</p><p>где $t_b$ и $t_c$ — также скалярные параметры.</p><p>Точка $A_2$ симметрична точке $A_1$ относительно середины $M_a$ стороны $BC$. Это означает, что $M_a$ является серединой отрезка $A_1A_2$. В векторной форме это условие записывается как $\vec{m}_a = \frac{\vec{a_1}+\vec{a_2}}{2}$. Отсюда можно выразить радиус-вектор точки $A_2$:</p><p>$\vec{a_2} = 2\vec{m}_a - \vec{a_1} = 2 \left( \frac{\vec{b}+\vec{c}}{2} \right) - \vec{a_1} = \vec{b}+\vec{c} - \vec{a_1}$</p><p>Подставим в это равенство выражение для $\vec{a_1}$:</p><p>$\vec{a_2} = \vec{b}+\vec{c} - ((1-t_a)\vec{b} + t_a\vec{c}) = (1 - (1-t_a))\vec{b} + (1-t_a)\vec{c} = t_a\vec{b} + (1-t_a)\vec{c}$</p><p>Аналогично найдем радиус-векторы для точек $B_2$ и $C_2$:</p><p>$\vec{b_2} = 2\vec{m}_b - \vec{b_1} = \vec{c}+\vec{a} - \vec{b_1} = \vec{c}+\vec{a} - ((1-t_b)\vec{c} + t_b\vec{a}) = t_b\vec{c} + (1-t_b)\vec{a}$</p><p>$\vec{c_2} = 2\vec{m}_c - \vec{c_1} = \vec{a}+\vec{b} - \vec{c_1} = \vec{a}+\vec{b} - ((1-t_c)\vec{a} + t_c\vec{b}) = t_c\vec{a} + (1-t_c)\vec{b}$</p><p>По условию задачи, точки $A_1, B_1, C_1$ лежат на одной прямой. Согласно теореме Менелая, это эквивалентно выполнению следующего равенства для отношений, в которых эти точки делят стороны треугольника (с учетом знака, то есть для ориентированных отрезков):</p><p>$\frac{\vec{BA_1}}{\vec{A_1C}} \cdot \frac{\vec{CB_1}}{\vec{B_1A}} \cdot \frac{\vec{AC_1}}{\vec{C_1B}} = 1$</p><p>Выразим эти отношения через параметры $t_a, t_b, t_c$.</p><p>Для точки $A_1$: $\vec{BA_1} = \vec{a_1} - \vec{b} = t_a(\vec{c}-\vec{b}) = t_a\vec{BC}$ и $\vec{A_1C} = \vec{c} - \vec{a_1} = (1-t_a)(\vec{c}-\vec{b}) = (1-t_a)\vec{BC}$. Отсюда $\frac{\vec{BA_1}}{\vec{A_1C}} = \frac{t_a}{1-t_a}$.</p><p>Аналогично, для точки $B_1$: $\frac{\vec{CB_1}}{\vec{B_1A}} = \frac{t_b}{1-t_b}$.</p><p>И для точки $C_1$: $\frac{\vec{AC_1}}{\vec{C_1B}} = \frac{t_c}{1-t_c}$.</p><p>Таким образом, условие коллинеарности точек $A_1, B_1, C_1$ принимает вид:</p><p>$\frac{t_a}{1-t_a} \cdot \frac{t_b}{1-t_b} \cdot \frac{t_c}{1-t_c} = 1$</p><p>Теперь проверим, выполняется ли условие теоремы Менелая для точек $A_2, B_2, C_2$. Для этого вычислим соответствующие отношения отрезков.</p><p>Для точки $A_2$: $\vec{BA_2} = \vec{a_2}-\vec{b} = (t_a-1)\vec{b} + (1-t_a)\vec{c} = (1-t_a)(\vec{c}-\vec{b}) = (1-t_a)\vec{BC}$ и $\vec{A_2C} = \vec{c}-\vec{a_2} = -t_a\vec{b} + t_a\vec{c} = t_a(\vec{c}-\vec{b}) = t_a\vec{BC}$. Отсюда $\frac{\vec{BA_2}}{\vec{A_2C}} = \frac{1-t_a}{t_a}$.</p><p>Аналогично, для точки $B_2$: $\frac{\vec{CB_2}}{\vec{B_2A}} = \frac{1-t_b}{t_b}$.</p><p>И для точки $C_2$: $\frac{\vec{AC_2}}{\vec{C_2B}} = \frac{1-t_c}{t_c}$.</p><p>Теперь найдем произведение этих отношений:</p><p>$\frac{\vec{BA_2}}{\vec{A_2C}} \cdot \frac{\vec{CB_2}}{\vec{B_2A}} \cdot \frac{\vec{AC_2}}{\vec{C_2B}} = \frac{1-t_a}{t_a} \cdot \frac{1-t_b}{t_b} \cdot \frac{1-t_c}{t_c}$</p><p>Это произведение является обратным к произведению для точек $A_1, B_1, C_1$:</p><p>$\frac{1-t_a}{t_a} \cdot \frac{1-t_b}{t_b} \cdot \frac{1-t_c}{t_c} = \left( \frac{t_a}{1-t_a} \cdot \frac{t_b}{1-t_b} \cdot \frac{t_c}{1-t_c} \right)^{-1}$</p><p>Поскольку мы установили, что произведение для коллинеарных точек $A_1, B_1, C_1$ равно 1, то и для $A_2, B_2, C_2$ получаем:</p><p>$\left(1\right)^{-1} = 1$</p><p>Так как для точек $A_2, B_2, C_2$ выполняется условие теоремы Менелая, то по обратной теореме Менелая эти точки лежат на одной прямой. Что и требовалось доказать.</p><p>Ответ: Точки $A_2, B_2, C_2$ также лежат на одной прямой.</p>" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "745269" "type" => "task" ] "previous" => array:2 [ "refs" => "745267" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1112 #items: array:1 [ 0 => App\Models\Book {#1049} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1042 #items: array:1 [ 0 => App\Models\BookPage {#1116 #connection: "mysql" #table: "book_pages" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 934000 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "218" "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/page-218" "field_display_title" => "218" "field_folder" => "1" "field_image_name" => "218" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1117 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1040 #items: array:1 [ 0 => App\Models\Book {#1049} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "edition_groups" => Illuminate\Database\Eloquent\Collection {#1125 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1147 #items: array:1 [ 0 => App\Models\Element {#1146 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1108264 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1148 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1108264 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1148 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "934001" "type" => "book_page" ] "previous" => array:2 [ "refs" => "933999" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1232 #items: array:7 [ 0 => App\Models\Task {#1246 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 745266 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "218" "field_page_end" => null "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/00-851" "field_display_title" => "851" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1247 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1248 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1249 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1250 …2} "content" => Illuminate\Database\Eloquent\Collection {#1252 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1261 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 745266 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "218" "field_page_end" => null "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/00-851" "field_display_title" => "851" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1247 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1248 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1249 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1250 …2} "content" => Illuminate\Database\Eloquent\Collection {#1252 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1261 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Task {#1253 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 745267 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "218" "field_page_end" => null "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/00-852" "field_display_title" => "852" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1251 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1259 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1258 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1260 …2} "content" => Illuminate\Database\Eloquent\Collection {#1254 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1277 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 745267 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "218" "field_page_end" => null "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/00-852" "field_display_title" => "852" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1251 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1259 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1258 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1260 …2} "content" => Illuminate\Database\Eloquent\Collection {#1254 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1277 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Task {#1257 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 745268 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "218" "field_page_end" => null "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/00-853" "field_display_title" => "853" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1256 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1275 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1274 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1276 …2} "content" => Illuminate\Database\Eloquent\Collection {#1272 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1255 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 745268 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "218" "field_page_end" => null "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/00-853" "field_display_title" => "853" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1256 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1275 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1274 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1276 …2} "content" => Illuminate\Database\Eloquent\Collection {#1272 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1255 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 3 => App\Models\Task {#1273 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 745269 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "218" "field_page_end" => null …19 ] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 4 => App\Models\Task {#1292 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 5 => App\Models\Task {#1296 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 6 => App\Models\Task {#1312 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } ] #original: array:21 [ "id" => 934000 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "218" "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/page-218" "field_display_title" => "218" "field_folder" => "1" "field_image_name" => "218" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1117} "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1040} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "edition_groups" => Illuminate\Database\Eloquent\Collection {#1125} "content" => Illuminate\Database\Eloquent\Collection {#1147} "next" => array:2 [ "refs" => "934001" "type" => "book_page" ] "previous" => array:2 [ "refs" => "933999" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1232} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 745268 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "218" "field_page_end" => null "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/00-853" "field_display_title" => "853" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1035} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1046} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1121} "content" => Illuminate\Database\Eloquent\Collection {#1038} "next" => array:2 [ "refs" => "745269" "type" => "task" ] "previous" => array:2 [ "refs" => "745267" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1112} "page" => array:2 [ "refs" => "934000" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] }
№853 (с. 218)
Условие. №853 (с. 218)
скриншот условия
853. На сторонах ВС, СА и АВ треугольника АВС или их продолжениях отмечены соответственно точки А₁, В₁ и С₁, лежащие на одной прямой. Докажите, что точки А₂, В₂ и С₂, симметричные соответственно точкам А₁, В₁ и С₁ относительно середин сторон ВС, СА и АВ, также лежат на одной прямой.
Решение 2. №853 (с. 218)
Решение 3. №853 (с. 218)
Решение 6. №853 (с. 218)
Для решения задачи воспользуемся векторным методом в сочетании с теоремой Менелая.
Пусть $O$ — произвольная точка плоскости (начало координат). Обозначим радиус-векторы вершин треугольника $A, B, C$ как $\vec{a}, \vec{b}, \vec{c}$ соответственно. Пусть $M_a, M_b, M_c$ — середины сторон $BC, CA, AB$. Их радиус-векторы равны:
$\vec{m}_a = \frac{\vec{b}+\vec{c}}{2}$, $\vec{m}_b = \frac{\vec{c}+\vec{a}}{2}$, $\vec{m}_c = \frac{\vec{a}+\vec{b}}{2}$
Точка $A_1$ лежит на прямой $BC$, поэтому ее радиус-вектор $\vec{a_1}$ можно представить как аффинную комбинацию векторов $\vec{b}$ и $\vec{c}$: $\vec{a_1} = (1-t_a)\vec{b} + t_a\vec{c}$ для некоторого скалярного параметра $t_a$. Аналогично для точек $B_1$ на прямой $CA$ и $C_1$ на прямой $AB$:
$\vec{b_1} = (1-t_b)\vec{c} + t_b\vec{a}$
$\vec{c_1} = (1-t_c)\vec{a} + t_c\vec{b}$
где $t_b$ и $t_c$ — также скалярные параметры.
Точка $A_2$ симметрична точке $A_1$ относительно середины $M_a$ стороны $BC$. Это означает, что $M_a$ является серединой отрезка $A_1A_2$. В векторной форме это условие записывается как $\vec{m}_a = \frac{\vec{a_1}+\vec{a_2}}{2}$. Отсюда можно выразить радиус-вектор точки $A_2$:
$\vec{a_2} = 2\vec{m}_a - \vec{a_1} = 2 \left( \frac{\vec{b}+\vec{c}}{2} \right) - \vec{a_1} = \vec{b}+\vec{c} - \vec{a_1}$
Подставим в это равенство выражение для $\vec{a_1}$:
$\vec{a_2} = \vec{b}+\vec{c} - ((1-t_a)\vec{b} + t_a\vec{c}) = (1 - (1-t_a))\vec{b} + (1-t_a)\vec{c} = t_a\vec{b} + (1-t_a)\vec{c}$
Аналогично найдем радиус-векторы для точек $B_2$ и $C_2$:
$\vec{b_2} = 2\vec{m}_b - \vec{b_1} = \vec{c}+\vec{a} - \vec{b_1} = \vec{c}+\vec{a} - ((1-t_b)\vec{c} + t_b\vec{a}) = t_b\vec{c} + (1-t_b)\vec{a}$
$\vec{c_2} = 2\vec{m}_c - \vec{c_1} = \vec{a}+\vec{b} - \vec{c_1} = \vec{a}+\vec{b} - ((1-t_c)\vec{a} + t_c\vec{b}) = t_c\vec{a} + (1-t_c)\vec{b}$
По условию задачи, точки $A_1, B_1, C_1$ лежат на одной прямой. Согласно теореме Менелая, это эквивалентно выполнению следующего равенства для отношений, в которых эти точки делят стороны треугольника (с учетом знака, то есть для ориентированных отрезков):
$\frac{\vec{BA_1}}{\vec{A_1C}} \cdot \frac{\vec{CB_1}}{\vec{B_1A}} \cdot \frac{\vec{AC_1}}{\vec{C_1B}} = 1$
Выразим эти отношения через параметры $t_a, t_b, t_c$.
Для точки $A_1$: $\vec{BA_1} = \vec{a_1} - \vec{b} = t_a(\vec{c}-\vec{b}) = t_a\vec{BC}$ и $\vec{A_1C} = \vec{c} - \vec{a_1} = (1-t_a)(\vec{c}-\vec{b}) = (1-t_a)\vec{BC}$. Отсюда $\frac{\vec{BA_1}}{\vec{A_1C}} = \frac{t_a}{1-t_a}$.
Аналогично, для точки $B_1$: $\frac{\vec{CB_1}}{\vec{B_1A}} = \frac{t_b}{1-t_b}$.
И для точки $C_1$: $\frac{\vec{AC_1}}{\vec{C_1B}} = \frac{t_c}{1-t_c}$.
Таким образом, условие коллинеарности точек $A_1, B_1, C_1$ принимает вид:
$\frac{t_a}{1-t_a} \cdot \frac{t_b}{1-t_b} \cdot \frac{t_c}{1-t_c} = 1$
Теперь проверим, выполняется ли условие теоремы Менелая для точек $A_2, B_2, C_2$. Для этого вычислим соответствующие отношения отрезков.
Для точки $A_2$: $\vec{BA_2} = \vec{a_2}-\vec{b} = (t_a-1)\vec{b} + (1-t_a)\vec{c} = (1-t_a)(\vec{c}-\vec{b}) = (1-t_a)\vec{BC}$ и $\vec{A_2C} = \vec{c}-\vec{a_2} = -t_a\vec{b} + t_a\vec{c} = t_a(\vec{c}-\vec{b}) = t_a\vec{BC}$. Отсюда $\frac{\vec{BA_2}}{\vec{A_2C}} = \frac{1-t_a}{t_a}$.
Аналогично, для точки $B_2$: $\frac{\vec{CB_2}}{\vec{B_2A}} = \frac{1-t_b}{t_b}$.
И для точки $C_2$: $\frac{\vec{AC_2}}{\vec{C_2B}} = \frac{1-t_c}{t_c}$.
Теперь найдем произведение этих отношений:
$\frac{\vec{BA_2}}{\vec{A_2C}} \cdot \frac{\vec{CB_2}}{\vec{B_2A}} \cdot \frac{\vec{AC_2}}{\vec{C_2B}} = \frac{1-t_a}{t_a} \cdot \frac{1-t_b}{t_b} \cdot \frac{1-t_c}{t_c}$
Это произведение является обратным к произведению для точек $A_1, B_1, C_1$:
$\frac{1-t_a}{t_a} \cdot \frac{1-t_b}{t_b} \cdot \frac{1-t_c}{t_c} = \left( \frac{t_a}{1-t_a} \cdot \frac{t_b}{1-t_b} \cdot \frac{t_c}{1-t_c} \right)^{-1}$
Поскольку мы установили, что произведение для коллинеарных точек $A_1, B_1, C_1$ равно 1, то и для $A_2, B_2, C_2$ получаем:
$\left(1\right)^{-1} = 1$
Так как для точек $A_2, B_2, C_2$ выполняется условие теоремы Менелая, то по обратной теореме Менелая эти точки лежат на одной прямой. Что и требовалось доказать.
Ответ: Точки $A_2, B_2, C_2$ также лежат на одной прямой.
Другие задания:
Помогло решение? Оставьте отзыв в комментариях ниже.
Присоединяйтесь к Телеграм-группе @top_gdz
ПрисоединитьсяМы подготовили для вас ответ c подробным объяснением домашего задания по геометрии за 10-11 класс, для упражнения номер 853 расположенного на странице 218 к учебнику 2019 года издания для учащихся школ и гимназий.
Теперь на нашем сайте ГДЗ.ТОП вы всегда легко и бесплатно найдёте условие с правильным ответом на вопрос «Как решить ДЗ» и «Как сделать» задание по геометрии к упражнению №853 (с. 218), авторов: Атанасян (Левон Сергеевич), Бутузов (Валентин Фёдорович), Кадомцев (Сергей Борисович), Позняк (Эдуард Генрихович), Киселёва (Людмила Сергеевна), ФГОС (старый) базовый и углублённый уровень обучения учебного пособия издательства Просвещение.