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Номер 862, страница 219 - 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 862, страница 219.
App\Models\Task {#1024 // 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" => 745277 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "219" "field_page_end" => null "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/00-862" "field_display_title" => "862" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#331 #items: array:1 [ 0 => App\Models\Term {#1029 #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 {#1026 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#342 #items: array:1 [ 0 => App\Models\Branch {#1023 #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 {#333 #items: array:1 [ 0 => App\Models\Term {#1033 #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 {#1030 #items: array:1 [ 0 => App\Models\Book {#1032 #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 {#1034 #items: array:1 [ 0 => App\Models\Term {#1035 #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 {#1036 #items: array:2 [ 0 => App\Models\Term {#1037 #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 {#1038 #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 {#1039 #items: array:1 [ 0 => App\Models\Term {#1040 #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 {#1041 #items: array:5 [ 0 => App\Models\Term {#1042 #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 {#1043 #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 {#1044 #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 {#1045 #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 {#1046 #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 {#1047 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1048 #items: array:1 [ 0 => App\Models\Term {#1049 #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 {#1050 #items: array:1 [ 0 => App\Models\Term {#1051 #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 {#1052 #items: array:1 [ 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" => 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 {#1054 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1055 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1056 #items: array:1 [ 0 => App\Models\Term {#1057 #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 {#1058 #items: array:1 [ 0 => 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: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 {#1060 #items: array:1 [ 0 => 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: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 {#1062 #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 {#1063 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1064 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1065 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/geometrija/atanasjan-uchebnik" "field_cover_color" => 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" => 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 {#1034} "field_class" => Illuminate\Database\Eloquent\Collection {#1036} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1039} "field_author" => Illuminate\Database\Eloquent\Collection {#1041} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1047} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1048} "field_country" => Illuminate\Database\Eloquent\Collection {#1050} "field_city" => Illuminate\Database\Eloquent\Collection {#1052} "field_series" => Illuminate\Database\Eloquent\Collection {#1054} "field_umk" => Illuminate\Database\Eloquent\Collection {#1055} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1056} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1058} "field_publication_number" => "11" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1060} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1062} "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 {#1063} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1064} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1065} "field_url" => "/10-klass/geometrija/atanasjan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1066} "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 {#1068 #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 {#333} "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 {#1030} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1068} ] #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 {#1069 #items: array:2 [ 0 => App\Models\Branch {#1023} 1 => App\Models\Branch {#1070 #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 {#1071 #items: array:1 [ 0 => App\Models\Term {#1072 #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 {#1073 #items: array:1 [ 0 => App\Models\Book {#1032} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1074 #items: array:1 [ 0 => App\Models\Branch {#1075 #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 {#1076 …2} "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 {#1077 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1078 …2} ] #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 {#1076 …2} "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 {#1077 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1078 …2} ] #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" => 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 {#1071} "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 {#1073} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1074} ] #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 {#1079 #items: array:4 [ 0 => App\Models\Element {#1080 #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" => 940271 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1081 #items: array:1 [ 0 => App\Models\Edition {#1082 #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 {#1083 …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 {#1085 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1086 …2} "field_process_formula" => "wiris" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1087 …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 {#1083 …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 {#1085 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1086 …2} "field_process_formula" => "wiris" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1087 …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" => "745277" "type" => "task" ] "text" => """ <p><strong>862.</strong> На сторонах АВ, ВС и СА треугольника АВС (либо на одной из сторон и продолжениях двух других сторон) отмечены соответственно точки С₁, А₁ и В₁. Докажите, что прямые АА₁, ВВ₁ и СС₁ пересекаются в одной точке тогда и только тогда, когда:</p><figure><figure>\n <img src="/media/geometrija_10/atanasjan-u23/0-00/862.webp?ts=1745836262" alt="Рисунок" loading="lazy" width="827" height="179">\n </figure>\n </figure><p>б) для любой точки О, не лежащей на прямых АВ, ВС и СА, выполняется равенство</p><figure><figure>\n <img src="/media/geometrija_10/atanasjan-u23/0-00/862_0.webp?ts=1745836262" alt="Рисунок" loading="lazy" width="758" height="179">\n </figure>\n <figcaption> </figcaption></figure> """ "img" => array:1 [ 0 => array:5 [ "name" => "862-1.jpg" "alt" => null "width" => "1542" "height" => 518 "path" => "/media/geometrija_10/atanasjan-u23/0-00/862-1.webp?ts=1739436954" ] ] ] #original: array:7 [ "id" => 940271 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1081} "task" => array:2 [ "refs" => "745277" "type" => "task" ] "text" => """ <p><strong>862.</strong> На сторонах АВ, ВС и СА треугольника АВС (либо на одной из сторон и продолжениях двух других сторон) отмечены соответственно точки С₁, А₁ и В₁. Докажите, что прямые АА₁, ВВ₁ и СС₁ пересекаются в одной точке тогда и только тогда, когда:</p><figure><figure>\n <img src="/media/geometrija_10/atanasjan-u23/0-00/862.webp?ts=1745836262" alt="Рисунок" loading="lazy" width="827" height="179">\n </figure>\n </figure><p>б) для любой точки О, не лежащей на прямых АВ, ВС и СА, выполняется равенство</p><figure><figure>\n <img src="/media/geometrija_10/atanasjan-u23/0-00/862_0.webp?ts=1745836262" alt="Рисунок" loading="lazy" width="758" height="179">\n </figure>\n <figcaption> </figcaption></figure> """ "img" => array:1 [ 0 => array:5 [ "name" => "862-1.jpg" "alt" => null "width" => "1542" "height" => 518 "path" => "/media/geometrija_10/atanasjan-u23/0-00/862-1.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 {#1088 #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" => 942823 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1089 #items: array:1 [ 0 => App\Models\Edition {#1090 #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 {#1093 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1094 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1095 …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 {#1093 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1094 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1095 …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" => "745277" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "862-1.png" "alt" => null "width" => "694" "height" => 4949 "path" => "/media/geometrija_10/atanasjan-u23/2-00/862-1.webp?ts=1739441758" ] 1 => array:5 [ "name" => "862-2.png" "alt" => null "width" => "694" "height" => 4987 "path" => "/media/geometrija_10/atanasjan-u23/2-00/862-2.webp?ts=1739441758" ] ] ] #original: array:6 [ "id" => 942823 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1089} "task" => array:2 [ "refs" => "745277" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "862-1.png" "alt" => null "width" => "694" "height" => 4949 "path" => "/media/geometrija_10/atanasjan-u23/2-00/862-1.webp?ts=1739441758" ] 1 => array:5 [ "name" => "862-2.png" "alt" => null "width" => "694" "height" => 4987 "path" => "/media/geometrija_10/atanasjan-u23/2-00/862-2.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 {#1096 #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" => 942561 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1097 #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" => 2318 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 3" "field_order" => "4" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1099 …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 {#1101 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1102 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1103 …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 {#1099 …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 {#1101 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1102 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1103 …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" => "745277" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "862-1.jpg" "alt" => null "width" => "1399" "height" => 1397 "path" => "/media/geometrija_10/atanasjan-u23/3-00/862-1.webp?ts=1739437267" ] 1 => array:5 [ "name" => "862-2.jpg" "alt" => null "width" => "1399" "height" => 1288 "path" => "/media/geometrija_10/atanasjan-u23/3-00/862-2.webp?ts=1739437267" ] ] ] #original: array:6 [ "id" => 942561 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1097} "task" => array:2 [ "refs" => "745277" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "862-1.jpg" "alt" => null "width" => "1399" "height" => 1397 "path" => "/media/geometrija_10/atanasjan-u23/3-00/862-1.webp?ts=1739437267" ] 1 => array:5 [ "name" => "862-2.jpg" "alt" => null "width" => "1399" "height" => 1288 "path" => "/media/geometrija_10/atanasjan-u23/3-00/862-2.webp?ts=1739437267" ] ] ] #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 {#1104 #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" => 1352590 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1105 #items: array:1 [ 0 => App\Models\Edition {#1106 #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 {#1107 …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 {#1109 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1110 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1111 …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 {#1107 …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 {#1109 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1110 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1111 …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" => "745277" "type" => "task" ] "text" => "<p>Это утверждение известно как тригонометрическая форма теоремы Чевы. Мы докажем оба пункта, показав эквивалентность данных условий стандартной теореме Чевы.</p><p>Стандартная теорема Чевы утверждает, что прямые $AA_1$, $BB_1$ и $CC_1$ пересекаются в одной точке тогда и только тогда, когда выполняется соотношение для длин отрезков:</p><p>$ \frac{AC_1}{C_1B} \cdot \frac{BA_1}{A_1C} \cdot \frac{CB_1}{B_1A} = 1 $</p><p>Это соотношение справедливо и в случае, когда точки $A_1, B_1, C_1$ лежат на продолжениях сторон (при использовании направленных отрезков, но для синусной формы это не требует отдельных оговорок, так как $\sin(180^\circ - \alpha) = \sin(\alpha)$).</p><p><strong>a)</strong> Докажем, что прямые $AA_1, BB_1, CC_1$ пересекаются в одной точке тогда и только тогда, когда $ \frac{\sin \angle ACC_1}{\sin \angle C_1CB} \cdot \frac{\sin \angle BAA_1}{\sin \angle A_1AC} \cdot \frac{\sin \angle CBB_1}{\sin \angle B_1BA} = 1 $.</p><p>Для доказательства мы установим эквивалентность этого условия и стандартной теоремы Чевы. Рассмотрим соотношения отрезков на сторонах треугольника.</p><p>Рассмотрим отношение $ \frac{AC_1}{C_1B} $. Применим теорему синусов к треугольникам $ACC_1$ и $BCC_1$.</p><p>В $\triangle ACC_1$: $ \frac{AC_1}{\sin \angle ACC_1} = \frac{CC_1}{\sin \angle A} $. Отсюда $ AC_1 = \frac{CC_1 \cdot \sin \angle ACC_1}{\sin \angle A} $.</p><p>В $\triangle BCC_1$: $ \frac{C_1B}{\sin \angle C_1CB} = \frac{CC_1}{\sin \angle B} $. Отсюда $ C_1B = \frac{CC_1 \cdot \sin \angle C_1CB}{\sin \angle B} $.</p><p>Разделив одно выражение на другое, получим:</p><p>$ \frac{AC_1}{C_1B} = \frac{\sin \angle ACC_1}{\sin \angle C_1CB} \cdot \frac{\sin \angle B}{\sin \angle A} $</p><p>Аналогично, для двух других отношений отрезков:</p><p>Для $ \frac{BA_1}{A_1C} $, применяя теорему синусов к $\triangle ABA_1$ и $\triangle ACA_1$:</p><p>$ \frac{BA_1}{A_1C} = \frac{\sin \angle BAA_1}{\sin \angle A_1AC} \cdot \frac{\sin \angle C}{\sin \angle B} $</p><p>Для $ \frac{CB_1}{B_1A} $, применяя теорему синусов к $\triangle BCB_1$ и $\triangle BAB_1$:</p><p>$ \frac{CB_1}{B_1A} = \frac{\sin \angle CBB_1}{\sin \angle B_1BA} \cdot \frac{\sin \angle A}{\sin \angle C} $</p><p>Теперь перемножим эти три равенства:</p><p>$ \frac{AC_1}{C_1B} \cdot \frac{BA_1}{A_1C} \cdot \frac{CB_1}{B_1A} = \left(\frac{\sin \angle ACC_1}{\sin \angle C_1CB} \cdot \frac{\sin \angle B}{\sin \angle A}\right) \cdot \left(\frac{\sin \angle BAA_1}{\sin \angle A_1AC} \cdot \frac{\sin \angle C}{\sin \angle B}\right) \cdot \left(\frac{\sin \angle CBB_1}{\sin \angle B_1BA} \cdot \frac{\sin \angle A}{\sin \angle C}\right) $</p><p>Сгруппируем множители:</p><p>$ \frac{AC_1}{C_1B} \cdot \frac{BA_1}{A_1C} \cdot \frac{CB_1}{B_1A} = \left(\frac{\sin \angle ACC_1}{\sin \angle C_1CB} \cdot \frac{\sin \angle BAA_1}{\sin \angle A_1AC} \cdot \frac{\sin \angle CBB_1}{\sin \angle B_1BA}\right) \cdot \left(\frac{\sin \angle B \cdot \sin \angle C \cdot \sin \angle A}{\sin \angle A \cdot \sin \angle B \cdot \sin \angle C}\right) $</p><p>Вторая скобка равна 1, следовательно:</p><p>$ \frac{AC_1}{C_1B} \cdot \frac{BA_1}{A_1C} \cdot \frac{CB_1}{B_1A} = \frac{\sin \angle ACC_1}{\sin \angle C_1CB} \cdot \frac{\sin \angle BAA_1}{\sin \angle A_1AC} \cdot \frac{\sin \angle CBB_1}{\sin \angle B_1BA} $</p><p>Из этого равенства следует, что условие $ \frac{AC_1}{C_1B} \cdot \frac{BA_1}{A_1C} \cdot \frac{CB_1}{B_1A} = 1 $ (стандартная теорема Чевы) эквивалентно условию $ \frac{\sin \angle ACC_1}{\sin \angle C_1CB} \cdot \frac{\sin \angle BAA_1}{\sin \angle A_1AC} \cdot \frac{\sin \angle CBB_1}{\sin \angle B_1BA} = 1 $.</p><p>Поскольку стандартная теорема Чевы является необходимым и достаточным условием пересечения прямых $AA_1, BB_1, CC_1$ в одной точке, то и данное в пункте а) условие также является необходимым и достаточным.</p><p>Ответ: Утверждение доказано.</p><p><strong>б)</strong> Докажем, что прямые $AA_1, BB_1, CC_1$ пересекаются в одной точке тогда и только тогда, когда для любой точки $O$, не лежащей на прямых $AB, BC$ и $CA$, выполняется равенство $ \frac{\sin \angle AOC_1}{\sin \angle C_1OB} \cdot \frac{\sin \angle BOA_1}{\sin \angle A_1OC} \cdot \frac{\sin \angle COB_1}{\sin \angle B_1OA} = 1 $.</p><p>Как и в пункте а), докажем эквивалентность этого условия и стандартной теоремы Чевы. Возьмем произвольную точку $O$, не лежащую на сторонах треугольника (или их продолжениях).</p><p>Рассмотрим отношение отрезков $ \frac{AC_1}{C_1B} $. Применим теорему синусов к треугольникам $OAC_1$ и $OBC_1$.</p><p>В $\triangle OAC_1$: $ \frac{AC_1}{\sin \angle AOC_1} = \frac{OA}{\sin \angle OC_1A} $.</p><p>В $\triangle OBC_1$: $ \frac{C_1B}{\sin \angle C_1OB} = \frac{OB}{\sin \angle OC_1B} $.</p><p>Поскольку точки $A, C_1, B$ лежат на одной прямой, углы $\angle OC_1A$ и $\angle OC_1B$ являются смежными, следовательно, $\sin \angle OC_1A = \sin \angle OC_1B$. Разделив первое уравнение на второе, получим:</p><p>$ \frac{AC_1}{C_1B} \cdot \frac{\sin \angle C_1OB}{\sin \angle AOC_1} = \frac{OA}{OB} \implies \frac{AC_1}{C_1B} = \frac{\sin \angle AOC_1}{\sin \angle C_1OB} \cdot \frac{OA}{OB} $</p><p>Аналогично, для двух других отношений отрезков:</p><p>Для $ \frac{BA_1}{A_1C} $, применяя теорему синусов к $\triangle OBA_1$ и $\triangle OCA_1$:</p><p>$ \frac{BA_1}{A_1C} = \frac{\sin \angle BOA_1}{\sin \angle A_1OC} \cdot \frac{OB}{OC} $</p><p>Для $ \frac{CB_1}{B_1A} $, применяя теорему синусов к $\triangle OCB_1$ и $\triangle OAB_1$:</p><p>$ \frac{CB_1}{B_1A} = \frac{\sin \angle COB_1}{\sin \angle B_1OA} \cdot \frac{OC}{OA} $</p><p>Теперь перемножим эти три равенства:</p><p>$ \frac{AC_1}{C_1B} \cdot \frac{BA_1}{A_1C} \cdot \frac{CB_1}{B_1A} = \left(\frac{\sin \angle AOC_1}{\sin \angle C_1OB} \cdot \frac{OA}{OB}\right) \cdot \left(\frac{\sin \angle BOA_1}{\sin \angle A_1OC} \cdot \frac{OB}{OC}\right) \cdot \left(\frac{\sin \angle COB_1}{\sin \angle B_1OA} \cdot \frac{OC}{OA}\right) $</p><p>Сгруппируем множители:</p><p>$ \frac{AC_1}{C_1B} \cdot \frac{BA_1}{A_1C} \cdot \frac{CB_1}{B_1A} = \left(\frac{\sin \angle AOC_1}{\sin \angle C_1OB} \cdot \frac{\sin \angle BOA_1}{\sin \angle A_1OC} \cdot \frac{\sin \angle COB_1}{\sin \angle B_1OA}\right) \cdot \left(\frac{OA}{OB} \cdot \frac{OB}{OC} \cdot \frac{OC}{OA}\right) $</p><p>Произведение во второй скобке равно 1. Таким образом, получаем:</p><p>$ \frac{AC_1}{C_1B} \cdot \frac{BA_1}{A_1C} \cdot \frac{CB_1}{B_1A} = \frac{\sin \angle AOC_1}{\sin \angle C_1OB} \cdot \frac{\sin \angle BOA_1}{\sin \angle A_1OC} \cdot \frac{\sin \angle COB_1}{\sin \angle B_1OA} $</p><p>Это равенство показывает, что условие стандартной теоремы Чевы $ \frac{AC_1}{C_1B} \cdot \frac{BA_1}{A_1C} \cdot \frac{CB_1}{B_1A} = 1 $ эквивалентно условию из пункта б). Следовательно, данное условие также является необходимым и достаточным для пересечения прямых $AA_1, BB_1, CC_1$ в одной точке.</p><p>Ответ: Утверждение доказано.</p>" ] #original: array:6 [ "id" => 1352590 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1105} "task" => array:2 [ "refs" => "745277" "type" => "task" ] "text" => "<p>Это утверждение известно как тригонометрическая форма теоремы Чевы. Мы докажем оба пункта, показав эквивалентность данных условий стандартной теореме Чевы.</p><p>Стандартная теорема Чевы утверждает, что прямые $AA_1$, $BB_1$ и $CC_1$ пересекаются в одной точке тогда и только тогда, когда выполняется соотношение для длин отрезков:</p><p>$ \frac{AC_1}{C_1B} \cdot \frac{BA_1}{A_1C} \cdot \frac{CB_1}{B_1A} = 1 $</p><p>Это соотношение справедливо и в случае, когда точки $A_1, B_1, C_1$ лежат на продолжениях сторон (при использовании направленных отрезков, но для синусной формы это не требует отдельных оговорок, так как $\sin(180^\circ - \alpha) = \sin(\alpha)$).</p><p><strong>a)</strong> Докажем, что прямые $AA_1, BB_1, CC_1$ пересекаются в одной точке тогда и только тогда, когда $ \frac{\sin \angle ACC_1}{\sin \angle C_1CB} \cdot \frac{\sin \angle BAA_1}{\sin \angle A_1AC} \cdot \frac{\sin \angle CBB_1}{\sin \angle B_1BA} = 1 $.</p><p>Для доказательства мы установим эквивалентность этого условия и стандартной теоремы Чевы. Рассмотрим соотношения отрезков на сторонах треугольника.</p><p>Рассмотрим отношение $ \frac{AC_1}{C_1B} $. Применим теорему синусов к треугольникам $ACC_1$ и $BCC_1$.</p><p>В $\triangle ACC_1$: $ \frac{AC_1}{\sin \angle ACC_1} = \frac{CC_1}{\sin \angle A} $. Отсюда $ AC_1 = \frac{CC_1 \cdot \sin \angle ACC_1}{\sin \angle A} $.</p><p>В $\triangle BCC_1$: $ \frac{C_1B}{\sin \angle C_1CB} = \frac{CC_1}{\sin \angle B} $. Отсюда $ C_1B = \frac{CC_1 \cdot \sin \angle C_1CB}{\sin \angle B} $.</p><p>Разделив одно выражение на другое, получим:</p><p>$ \frac{AC_1}{C_1B} = \frac{\sin \angle ACC_1}{\sin \angle C_1CB} \cdot \frac{\sin \angle B}{\sin \angle A} $</p><p>Аналогично, для двух других отношений отрезков:</p><p>Для $ \frac{BA_1}{A_1C} $, применяя теорему синусов к $\triangle ABA_1$ и $\triangle ACA_1$:</p><p>$ \frac{BA_1}{A_1C} = \frac{\sin \angle BAA_1}{\sin \angle A_1AC} \cdot \frac{\sin \angle C}{\sin \angle B} $</p><p>Для $ \frac{CB_1}{B_1A} $, применяя теорему синусов к $\triangle BCB_1$ и $\triangle BAB_1$:</p><p>$ \frac{CB_1}{B_1A} = \frac{\sin \angle CBB_1}{\sin \angle B_1BA} \cdot \frac{\sin \angle A}{\sin \angle C} $</p><p>Теперь перемножим эти три равенства:</p><p>$ \frac{AC_1}{C_1B} \cdot \frac{BA_1}{A_1C} \cdot \frac{CB_1}{B_1A} = \left(\frac{\sin \angle ACC_1}{\sin \angle C_1CB} \cdot \frac{\sin \angle B}{\sin \angle A}\right) \cdot \left(\frac{\sin \angle BAA_1}{\sin \angle A_1AC} \cdot \frac{\sin \angle C}{\sin \angle B}\right) \cdot \left(\frac{\sin \angle CBB_1}{\sin \angle B_1BA} \cdot \frac{\sin \angle A}{\sin \angle C}\right) $</p><p>Сгруппируем множители:</p><p>$ \frac{AC_1}{C_1B} \cdot \frac{BA_1}{A_1C} \cdot \frac{CB_1}{B_1A} = \left(\frac{\sin \angle ACC_1}{\sin \angle C_1CB} \cdot \frac{\sin \angle BAA_1}{\sin \angle A_1AC} \cdot \frac{\sin \angle CBB_1}{\sin \angle B_1BA}\right) \cdot \left(\frac{\sin \angle B \cdot \sin \angle C \cdot \sin \angle A}{\sin \angle A \cdot \sin \angle B \cdot \sin \angle C}\right) $</p><p>Вторая скобка равна 1, следовательно:</p><p>$ \frac{AC_1}{C_1B} \cdot \frac{BA_1}{A_1C} \cdot \frac{CB_1}{B_1A} = \frac{\sin \angle ACC_1}{\sin \angle C_1CB} \cdot \frac{\sin \angle BAA_1}{\sin \angle A_1AC} \cdot \frac{\sin \angle CBB_1}{\sin \angle B_1BA} $</p><p>Из этого равенства следует, что условие $ \frac{AC_1}{C_1B} \cdot \frac{BA_1}{A_1C} \cdot \frac{CB_1}{B_1A} = 1 $ (стандартная теорема Чевы) эквивалентно условию $ \frac{\sin \angle ACC_1}{\sin \angle C_1CB} \cdot \frac{\sin \angle BAA_1}{\sin \angle A_1AC} \cdot \frac{\sin \angle CBB_1}{\sin \angle B_1BA} = 1 $.</p><p>Поскольку стандартная теорема Чевы является необходимым и достаточным условием пересечения прямых $AA_1, BB_1, CC_1$ в одной точке, то и данное в пункте а) условие также является необходимым и достаточным.</p><p>Ответ: Утверждение доказано.</p><p><strong>б)</strong> Докажем, что прямые $AA_1, BB_1, CC_1$ пересекаются в одной точке тогда и только тогда, когда для любой точки $O$, не лежащей на прямых $AB, BC$ и $CA$, выполняется равенство $ \frac{\sin \angle AOC_1}{\sin \angle C_1OB} \cdot \frac{\sin \angle BOA_1}{\sin \angle A_1OC} \cdot \frac{\sin \angle COB_1}{\sin \angle B_1OA} = 1 $.</p><p>Как и в пункте а), докажем эквивалентность этого условия и стандартной теоремы Чевы. Возьмем произвольную точку $O$, не лежащую на сторонах треугольника (или их продолжениях).</p><p>Рассмотрим отношение отрезков $ \frac{AC_1}{C_1B} $. Применим теорему синусов к треугольникам $OAC_1$ и $OBC_1$.</p><p>В $\triangle OAC_1$: $ \frac{AC_1}{\sin \angle AOC_1} = \frac{OA}{\sin \angle OC_1A} $.</p><p>В $\triangle OBC_1$: $ \frac{C_1B}{\sin \angle C_1OB} = \frac{OB}{\sin \angle OC_1B} $.</p><p>Поскольку точки $A, C_1, B$ лежат на одной прямой, углы $\angle OC_1A$ и $\angle OC_1B$ являются смежными, следовательно, $\sin \angle OC_1A = \sin \angle OC_1B$. Разделив первое уравнение на второе, получим:</p><p>$ \frac{AC_1}{C_1B} \cdot \frac{\sin \angle C_1OB}{\sin \angle AOC_1} = \frac{OA}{OB} \implies \frac{AC_1}{C_1B} = \frac{\sin \angle AOC_1}{\sin \angle C_1OB} \cdot \frac{OA}{OB} $</p><p>Аналогично, для двух других отношений отрезков:</p><p>Для $ \frac{BA_1}{A_1C} $, применяя теорему синусов к $\triangle OBA_1$ и $\triangle OCA_1$:</p><p>$ \frac{BA_1}{A_1C} = \frac{\sin \angle BOA_1}{\sin \angle A_1OC} \cdot \frac{OB}{OC} $</p><p>Для $ \frac{CB_1}{B_1A} $, применяя теорему синусов к $\triangle OCB_1$ и $\triangle OAB_1$:</p><p>$ \frac{CB_1}{B_1A} = \frac{\sin \angle COB_1}{\sin \angle B_1OA} \cdot \frac{OC}{OA} $</p><p>Теперь перемножим эти три равенства:</p><p>$ \frac{AC_1}{C_1B} \cdot \frac{BA_1}{A_1C} \cdot \frac{CB_1}{B_1A} = \left(\frac{\sin \angle AOC_1}{\sin \angle C_1OB} \cdot \frac{OA}{OB}\right) \cdot \left(\frac{\sin \angle BOA_1}{\sin \angle A_1OC} \cdot \frac{OB}{OC}\right) \cdot \left(\frac{\sin \angle COB_1}{\sin \angle B_1OA} \cdot \frac{OC}{OA}\right) $</p><p>Сгруппируем множители:</p><p>$ \frac{AC_1}{C_1B} \cdot \frac{BA_1}{A_1C} \cdot \frac{CB_1}{B_1A} = \left(\frac{\sin \angle AOC_1}{\sin \angle C_1OB} \cdot \frac{\sin \angle BOA_1}{\sin \angle A_1OC} \cdot \frac{\sin \angle COB_1}{\sin \angle B_1OA}\right) \cdot \left(\frac{OA}{OB} \cdot \frac{OB}{OC} \cdot \frac{OC}{OA}\right) $</p><p>Произведение во второй скобке равно 1. Таким образом, получаем:</p><p>$ \frac{AC_1}{C_1B} \cdot \frac{BA_1}{A_1C} \cdot \frac{CB_1}{B_1A} = \frac{\sin \angle AOC_1}{\sin \angle C_1OB} \cdot \frac{\sin \angle BOA_1}{\sin \angle A_1OC} \cdot \frac{\sin \angle COB_1}{\sin \angle B_1OA} $</p><p>Это равенство показывает, что условие стандартной теоремы Чевы $ \frac{AC_1}{C_1B} \cdot \frac{BA_1}{A_1C} \cdot \frac{CB_1}{B_1A} = 1 $ эквивалентно условию из пункта б). Следовательно, данное условие также является необходимым и достаточным для пересечения прямых $AA_1, BB_1, CC_1$ в одной точке.</p><p>Ответ: Утверждение доказано.</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" => "745278" "type" => "task" ] "previous" => array:2 [ "refs" => "745276" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1112 #items: array:1 [ 0 => App\Models\Book {#1032} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1292 #items: array:1 [ 0 => App\Models\BookPage {#1116 #connection: "mysql" #table: "book_pages" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: 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Illuminate\Database\Eloquent\Collection {#1152 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1153 …2} "field_url" => "/10-klass/geometrija/atanasjan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1154 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #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 {#1122 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1124 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1127 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1129 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1135 …2} "field_book_type" => 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№862 (с. 219)
Условие. №862 (с. 219)
скриншот условия
862. На сторонах АВ, ВС и СА треугольника АВС (либо на одной из сторон и продолжениях двух других сторон) отмечены соответственно точки С₁, А₁ и В₁. Докажите, что прямые АА₁, ВВ₁ и СС₁ пересекаются в одной точке тогда и только тогда, когда:
б) для любой точки О, не лежащей на прямых АВ, ВС и СА, выполняется равенство
Решение 2. №862 (с. 219)
Решение 3. №862 (с. 219)
Решение 6. №862 (с. 219)
Это утверждение известно как тригонометрическая форма теоремы Чевы. Мы докажем оба пункта, показав эквивалентность данных условий стандартной теореме Чевы.
Стандартная теорема Чевы утверждает, что прямые $AA_1$, $BB_1$ и $CC_1$ пересекаются в одной точке тогда и только тогда, когда выполняется соотношение для длин отрезков:
$ \frac{AC_1}{C_1B} \cdot \frac{BA_1}{A_1C} \cdot \frac{CB_1}{B_1A} = 1 $
Это соотношение справедливо и в случае, когда точки $A_1, B_1, C_1$ лежат на продолжениях сторон (при использовании направленных отрезков, но для синусной формы это не требует отдельных оговорок, так как $\sin(180^\circ - \alpha) = \sin(\alpha)$).
a) Докажем, что прямые $AA_1, BB_1, CC_1$ пересекаются в одной точке тогда и только тогда, когда $ \frac{\sin \angle ACC_1}{\sin \angle C_1CB} \cdot \frac{\sin \angle BAA_1}{\sin \angle A_1AC} \cdot \frac{\sin \angle CBB_1}{\sin \angle B_1BA} = 1 $.
Для доказательства мы установим эквивалентность этого условия и стандартной теоремы Чевы. Рассмотрим соотношения отрезков на сторонах треугольника.
Рассмотрим отношение $ \frac{AC_1}{C_1B} $. Применим теорему синусов к треугольникам $ACC_1$ и $BCC_1$.
В $\triangle ACC_1$: $ \frac{AC_1}{\sin \angle ACC_1} = \frac{CC_1}{\sin \angle A} $. Отсюда $ AC_1 = \frac{CC_1 \cdot \sin \angle ACC_1}{\sin \angle A} $.
В $\triangle BCC_1$: $ \frac{C_1B}{\sin \angle C_1CB} = \frac{CC_1}{\sin \angle B} $. Отсюда $ C_1B = \frac{CC_1 \cdot \sin \angle C_1CB}{\sin \angle B} $.
Разделив одно выражение на другое, получим:
$ \frac{AC_1}{C_1B} = \frac{\sin \angle ACC_1}{\sin \angle C_1CB} \cdot \frac{\sin \angle B}{\sin \angle A} $
Аналогично, для двух других отношений отрезков:
Для $ \frac{BA_1}{A_1C} $, применяя теорему синусов к $\triangle ABA_1$ и $\triangle ACA_1$:
$ \frac{BA_1}{A_1C} = \frac{\sin \angle BAA_1}{\sin \angle A_1AC} \cdot \frac{\sin \angle C}{\sin \angle B} $
Для $ \frac{CB_1}{B_1A} $, применяя теорему синусов к $\triangle BCB_1$ и $\triangle BAB_1$:
$ \frac{CB_1}{B_1A} = \frac{\sin \angle CBB_1}{\sin \angle B_1BA} \cdot \frac{\sin \angle A}{\sin \angle C} $
Теперь перемножим эти три равенства:
$ \frac{AC_1}{C_1B} \cdot \frac{BA_1}{A_1C} \cdot \frac{CB_1}{B_1A} = \left(\frac{\sin \angle ACC_1}{\sin \angle C_1CB} \cdot \frac{\sin \angle B}{\sin \angle A}\right) \cdot \left(\frac{\sin \angle BAA_1}{\sin \angle A_1AC} \cdot \frac{\sin \angle C}{\sin \angle B}\right) \cdot \left(\frac{\sin \angle CBB_1}{\sin \angle B_1BA} \cdot \frac{\sin \angle A}{\sin \angle C}\right) $
Сгруппируем множители:
$ \frac{AC_1}{C_1B} \cdot \frac{BA_1}{A_1C} \cdot \frac{CB_1}{B_1A} = \left(\frac{\sin \angle ACC_1}{\sin \angle C_1CB} \cdot \frac{\sin \angle BAA_1}{\sin \angle A_1AC} \cdot \frac{\sin \angle CBB_1}{\sin \angle B_1BA}\right) \cdot \left(\frac{\sin \angle B \cdot \sin \angle C \cdot \sin \angle A}{\sin \angle A \cdot \sin \angle B \cdot \sin \angle C}\right) $
Вторая скобка равна 1, следовательно:
$ \frac{AC_1}{C_1B} \cdot \frac{BA_1}{A_1C} \cdot \frac{CB_1}{B_1A} = \frac{\sin \angle ACC_1}{\sin \angle C_1CB} \cdot \frac{\sin \angle BAA_1}{\sin \angle A_1AC} \cdot \frac{\sin \angle CBB_1}{\sin \angle B_1BA} $
Из этого равенства следует, что условие $ \frac{AC_1}{C_1B} \cdot \frac{BA_1}{A_1C} \cdot \frac{CB_1}{B_1A} = 1 $ (стандартная теорема Чевы) эквивалентно условию $ \frac{\sin \angle ACC_1}{\sin \angle C_1CB} \cdot \frac{\sin \angle BAA_1}{\sin \angle A_1AC} \cdot \frac{\sin \angle CBB_1}{\sin \angle B_1BA} = 1 $.
Поскольку стандартная теорема Чевы является необходимым и достаточным условием пересечения прямых $AA_1, BB_1, CC_1$ в одной точке, то и данное в пункте а) условие также является необходимым и достаточным.
Ответ: Утверждение доказано.
б) Докажем, что прямые $AA_1, BB_1, CC_1$ пересекаются в одной точке тогда и только тогда, когда для любой точки $O$, не лежащей на прямых $AB, BC$ и $CA$, выполняется равенство $ \frac{\sin \angle AOC_1}{\sin \angle C_1OB} \cdot \frac{\sin \angle BOA_1}{\sin \angle A_1OC} \cdot \frac{\sin \angle COB_1}{\sin \angle B_1OA} = 1 $.
Как и в пункте а), докажем эквивалентность этого условия и стандартной теоремы Чевы. Возьмем произвольную точку $O$, не лежащую на сторонах треугольника (или их продолжениях).
Рассмотрим отношение отрезков $ \frac{AC_1}{C_1B} $. Применим теорему синусов к треугольникам $OAC_1$ и $OBC_1$.
В $\triangle OAC_1$: $ \frac{AC_1}{\sin \angle AOC_1} = \frac{OA}{\sin \angle OC_1A} $.
В $\triangle OBC_1$: $ \frac{C_1B}{\sin \angle C_1OB} = \frac{OB}{\sin \angle OC_1B} $.
Поскольку точки $A, C_1, B$ лежат на одной прямой, углы $\angle OC_1A$ и $\angle OC_1B$ являются смежными, следовательно, $\sin \angle OC_1A = \sin \angle OC_1B$. Разделив первое уравнение на второе, получим:
$ \frac{AC_1}{C_1B} \cdot \frac{\sin \angle C_1OB}{\sin \angle AOC_1} = \frac{OA}{OB} \implies \frac{AC_1}{C_1B} = \frac{\sin \angle AOC_1}{\sin \angle C_1OB} \cdot \frac{OA}{OB} $
Аналогично, для двух других отношений отрезков:
Для $ \frac{BA_1}{A_1C} $, применяя теорему синусов к $\triangle OBA_1$ и $\triangle OCA_1$:
$ \frac{BA_1}{A_1C} = \frac{\sin \angle BOA_1}{\sin \angle A_1OC} \cdot \frac{OB}{OC} $
Для $ \frac{CB_1}{B_1A} $, применяя теорему синусов к $\triangle OCB_1$ и $\triangle OAB_1$:
$ \frac{CB_1}{B_1A} = \frac{\sin \angle COB_1}{\sin \angle B_1OA} \cdot \frac{OC}{OA} $
Теперь перемножим эти три равенства:
$ \frac{AC_1}{C_1B} \cdot \frac{BA_1}{A_1C} \cdot \frac{CB_1}{B_1A} = \left(\frac{\sin \angle AOC_1}{\sin \angle C_1OB} \cdot \frac{OA}{OB}\right) \cdot \left(\frac{\sin \angle BOA_1}{\sin \angle A_1OC} \cdot \frac{OB}{OC}\right) \cdot \left(\frac{\sin \angle COB_1}{\sin \angle B_1OA} \cdot \frac{OC}{OA}\right) $
Сгруппируем множители:
$ \frac{AC_1}{C_1B} \cdot \frac{BA_1}{A_1C} \cdot \frac{CB_1}{B_1A} = \left(\frac{\sin \angle AOC_1}{\sin \angle C_1OB} \cdot \frac{\sin \angle BOA_1}{\sin \angle A_1OC} \cdot \frac{\sin \angle COB_1}{\sin \angle B_1OA}\right) \cdot \left(\frac{OA}{OB} \cdot \frac{OB}{OC} \cdot \frac{OC}{OA}\right) $
Произведение во второй скобке равно 1. Таким образом, получаем:
$ \frac{AC_1}{C_1B} \cdot \frac{BA_1}{A_1C} \cdot \frac{CB_1}{B_1A} = \frac{\sin \angle AOC_1}{\sin \angle C_1OB} \cdot \frac{\sin \angle BOA_1}{\sin \angle A_1OC} \cdot \frac{\sin \angle COB_1}{\sin \angle B_1OA} $
Это равенство показывает, что условие стандартной теоремы Чевы $ \frac{AC_1}{C_1B} \cdot \frac{BA_1}{A_1C} \cdot \frac{CB_1}{B_1A} = 1 $ эквивалентно условию из пункта б). Следовательно, данное условие также является необходимым и достаточным для пересечения прямых $AA_1, BB_1, CC_1$ в одной точке.
Ответ: Утверждение доказано.
Другие задания:
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