gdz.top
Номер 4, страница 6 - гдз по геометрии 10 класс рабочая тетрадь Глазков, Юдина
Авторы: Глазков Ю. А., Юдина И. И., Бутузов В. Ф.
Тип: рабочая тетрадь
Серия: мгу - школе
Издательство: Просвещение
Год издания: 2020 - 2026
Уровень обучения: базовый и углублённый
Цвет обложки: коричневый в сеточку
ISBN: 978-5-09-097573-5
Популярные ГДЗ в 10 классе
Аксиомы стереометрии - номер 4, страница 6.
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" => 738471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "6" "field_page_end" => null "field_url" => "/10-klass/geometrija/glazkov-tetrad/00-4" "field_display_title" => "4" "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" => 738452 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Аксиомы стереометрии" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#333 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "3" "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 {#1033 #items: array:1 [ 0 => App\Models\Book {#1030 #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" => 261 "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 {#1032 #items: array:1 [ 0 => App\Models\Term {#1034 #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 [ "field_accusative_case" => "геометрию" "field_creative_case" => "геометрией" "field_dative_case" => "геометрии" "field_genitive_case" => "геометрии" "field_nominative_case" => "геометрия" "field_prepositional_case" => "геометрии" ] "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 [ "field_accusative_case" => "геометрию" "field_creative_case" => "геометрией" "field_dative_case" => "геометрии" "field_genitive_case" => "геометрии" "field_nominative_case" => "геометрия" "field_prepositional_case" => "геометрии" ] "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 {#1035 #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" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ "field_accusative_case" => "десятый" "field_creative_case" => "десятым" "field_dative_case" => "десятому" "field_genitive_case" => "десятого" "field_nominative_case" => "десятый" "field_prepositional_case" => "десятом" ] "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 [ "field_accusative_case" => "десятый" "field_creative_case" => "десятым" "field_dative_case" => "десятому" "field_genitive_case" => "десятого" "field_nominative_case" => "десятый" "field_prepositional_case" => "десятом" ] "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 => "*" ] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1037 #items: array:1 [ 0 => 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" => 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 {#1039 #items: array:3 [ 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:12 [ "id" => 4121 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глазков" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Юрий" "field_patronymic" => "Александрович" "field_surname_rp" => null ] #original: array:12 [ "id" => 4121 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глазков" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Юрий" "field_patronymic" => "Александрович" "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 {#1041 #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" => 6432 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Юдина" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Ирина" "field_patronymic" => "Игоревна" "field_surname_rp" => null ] #original: array:12 [ "id" => 6432 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Юдина" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Ирина" "field_patronymic" => "Игоревна" "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 {#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" => 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 => "*" ] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1043 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1044 #items: array:1 [ 0 => 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:10 [ "id" => 6736 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "рабочая тетрадь" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "рабочую тетрадь" "field_creative_case" => "рабочей тетрадью" "field_dative_case" => "рабочей тетради" "field_genitive_case" => "рабочей тетради" "field_nominative_case" => "рабочая тетрадь" "field_prepositional_case" => "рабочей тетради" ] "field_plural_form" => "рабочие тетради" "field_short_name" => "рт" "field_short_name_foreign" => null "field_translit" => "rabochaya tetrad" ] #original: array:10 [ "id" => 6736 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "рабочая тетрадь" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "рабочую тетрадь" "field_creative_case" => "рабочей тетрадью" "field_dative_case" => "рабочей тетради" "field_genitive_case" => "рабочей тетради" "field_nominative_case" => "рабочая тетрадь" "field_prepositional_case" => "рабочей тетради" ] "field_plural_form" => "рабочие тетради" "field_short_name" => "рт" "field_short_name_foreign" => null "field_translit" => "rabochaya tetrad" ] #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 {#1046 #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:6 [ "id" => 9 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Россия" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "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 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "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 {#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:6 [ "id" => 15 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Москва" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "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 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "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 {#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:8 [ "id" => 4728 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "мгу - школе" "field_cases" => null "field_foreign_lang_series" => null "field_transcription_eng" => null "field_translit" => "mgu - shkole" ] #original: array:8 [ "id" => 4728 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "мгу - школе" "field_cases" => null "field_foreign_lang_series" => null "field_transcription_eng" => null "field_translit" => "mgu - shkole" ] #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_umk" => Illuminate\Database\Eloquent\Collection {#1052 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1053 #items: array:1 [ 0 => 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" => 41 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "базовый и углублённый" "field_cases" => array:6 [ "field_accusative_case" => "базовый и углубленный" "field_creative_case" => "базовым и углубленным" "field_dative_case" => "базовому и углубленному" "field_genitive_case" => "базового и углубленного" "field_nominative_case" => "базовый и углубленный" "field_prepositional_case" => "базовом и углубленном" ] "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 [ "field_accusative_case" => "базовый и углубленный" "field_creative_case" => "базовым и углубленным" "field_dative_case" => "базовому и углубленному" "field_genitive_case" => "базового и углубленного" "field_nominative_case" => "базовый и углубленный" "field_prepositional_case" => "базовом и углубленном" ] "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 {#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:10 [ "id" => 6706 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "ФГОС (старый)" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "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 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "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" => "4" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1057 #items: array:1 [ 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:6 [ "id" => 33 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "стереотипное" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "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 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "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 {#1059 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => "в сеточку" "field_publication_year" => "2020" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное пособие" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "110" "field_priority" => "14" "field_default_folder" => "/geometrija_10/glazkov-rt/" "field_isbn" => "978-5-09-097573-5" "field_cover" => array:1 [ 0 => "/media/geometrija_10/glazkov-rt/covers/cover1.webp?ts=1739295144" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/geometrija_10/glazkov-rt/covers/cover1.webp?ts=1739295144" "alt" => "" "width" => "1200" "height" => "1661" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1060 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1061 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1062 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/geometrija/glazkov-tetrad" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1063 #items: array:1 [ 0 => App\Models\Term {#1064 #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 [ "field_accusative_case" => "коричневый" "field_creative_case" => "коричневым" "field_dative_case" => "коричневому" "field_genitive_case" => "коричневого" "field_nominative_case" => "коричневый" "field_prepositional_case" => "коричневом" ] "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 [ "field_accusative_case" => "коричневый" "field_creative_case" => "коричневым" "field_dative_case" => "коричневому" "field_genitive_case" => "коричневого" "field_nominative_case" => "коричневый" "field_prepositional_case" => "коричневом" ] "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" => 261 "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 {#1032} "field_class" => Illuminate\Database\Eloquent\Collection {#1035} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1037} "field_author" => Illuminate\Database\Eloquent\Collection {#1039} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1043} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1044} "field_country" => Illuminate\Database\Eloquent\Collection {#1046} "field_city" => Illuminate\Database\Eloquent\Collection {#1048} "field_series" => Illuminate\Database\Eloquent\Collection {#1050} "field_umk" => Illuminate\Database\Eloquent\Collection {#1052} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1053} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1055} "field_publication_number" => "4" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1057} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1059} "field_under_the_edition_degree" => null "field_cover_description" => "в сеточку" "field_publication_year" => "2020" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное пособие" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "110" "field_priority" => "14" "field_default_folder" => "/geometrija_10/glazkov-rt/" "field_isbn" => "978-5-09-097573-5" "field_cover" => array:1 [ 0 => "/media/geometrija_10/glazkov-rt/covers/cover1.webp?ts=1739295144" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/geometrija_10/glazkov-rt/covers/cover1.webp?ts=1739295144" "alt" => "" "width" => "1200" "height" => "1661" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1060} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1061} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1062} "field_url" => "/10-klass/geometrija/glazkov-tetrad" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1063} "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 {#1065 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 738452 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Аксиомы стереометрии" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#333} "field_page_start" => "3" "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 {#1033} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1065} ] #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 {#1066 #items: array:1 [ 0 => App\Models\Branch {#1023} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1067 #items: array:3 [ 0 => App\Models\Element {#1068 #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" => 937085 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1069 #items: array:1 [ 0 => App\Models\Edition {#1070 #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" => 2220 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => 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:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "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" => "margarita" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1073 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1074 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1075 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 2220 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1071} "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" => "margarita" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1073} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1074} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1075} "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" => "738471" "type" => "task" ] "text" => "<p>На рисунке прямые $a$ и $b$ пересекаются в точке $P$. Докажите, что все прямые, не проходящие через точку $P$ и пересекающие прямые $a$ и $b$ в каких-то точках $X$ и $Y$, лежат в одной плоскости.</p><p><strong>Доказательство.</strong></p><p>По через пересекающиеся прямые $a$ и $b$ проходит некоторая плоскость $\alpha$, причем $X \in \alpha$ и $Y \in \alpha$, так как прямые $a$ и $b$.</p><p>Поэтому, согласно , прямая $XY$ лежит в плоскости $\alpha$. Итак, все рассматриваемые прямые лежат в .</p>" "img" => array:1 [ 0 => array:5 [ "name" => "4-1.jpg" "alt" => null "width" => "1677" "height" => 1042 "path" => "/media/geometrija_10/glazkov-rt/0-00/4-1.webp?ts=1739375698" ] ] ] #original: array:7 [ "id" => 937085 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1069} "task" => array:2 [ "refs" => "738471" "type" => "task" ] "text" => "<p>На рисунке прямые $a$ и $b$ пересекаются в точке $P$. Докажите, что все прямые, не проходящие через точку $P$ и пересекающие прямые $a$ и $b$ в каких-то точках $X$ и $Y$, лежат в одной плоскости.</p><p><strong>Доказательство.</strong></p><p>По через пересекающиеся прямые $a$ и $b$ проходит некоторая плоскость $\alpha$, причем $X \in \alpha$ и $Y \in \alpha$, так как прямые $a$ и $b$.</p><p>Поэтому, согласно , прямая $XY$ лежит в плоскости $\alpha$. Итак, все рассматриваемые прямые лежат в .</p>" "img" => array:1 [ 0 => array:5 [ "name" => "4-1.jpg" "alt" => null "width" => "1677" "height" => 1042 "path" => "/media/geometrija_10/glazkov-rt/0-00/4-1.webp?ts=1739375698" ] ] ] #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 {#1076 #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" => 937195 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1077 #items: array:1 [ 0 => App\Models\Edition {#1078 #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" => 2221 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1079 #items: array:1 [ 0 => App\Models\Term {#1080 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "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" => "margarita" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1081 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1082 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1083 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 2221 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1079} "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" => "margarita" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1081} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1082} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1083} "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" => "738471" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "4-1.png" "alt" => null "width" => "694" "height" => 2762 "path" => "/media/geometrija_10/glazkov-rt/1-00/4-1.webp?ts=1739375830" ] ] ] #original: array:6 [ "id" => 937195 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1077} "task" => array:2 [ "refs" => "738471" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "4-1.png" "alt" => null "width" => "694" "height" => 2762 "path" => "/media/geometrija_10/glazkov-rt/1-00/4-1.webp?ts=1739375830" ] ] ] #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 {#1084 #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" => 1434269 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1085 #items: array:1 [ 0 => App\Models\Edition {#1086 #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" => 5337 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1087 #items: array:1 [ 0 => App\Models\Term {#1088 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "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" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1089 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1090 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1091 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5337 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1087} "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" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1089} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1090} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1091} "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" => "738471" "type" => "task" ] "text" => "<p><strong>Доказательство.</strong></p> <p> Это утверждение является следствием основных аксиом и теорем стереометрии. Приведем развернутое доказательство. </p> <p> 1. Согласно теореме о существовании и единственности плоскости, проходящей через две пересекающиеся прямые, через прямые $a$ и $b$, которые пересекаются в точке $P$, можно провести единственную плоскость. Назовем эту плоскость $\alpha$. Таким образом, обе прямые, $a$ и $b$, целиком лежат в этой плоскости ($a \subset \alpha$ и $b \subset \alpha$). </p> <p> 2. Рассмотрим любую прямую (обозначим ее $c$), которая удовлетворяет условиям задачи: она не проходит через точку $P$, но пересекает прямую $a$ в некоторой точке $X$ и прямую $b$ в некоторой точке $Y$. </p> <p> 3. Так как точка $X$ лежит на прямой $a$ ($X \in a$), а прямая $a$ лежит в плоскости $\alpha$ ($a \subset \alpha$), то по определению принадлежности точка $X$ также лежит в плоскости $\alpha$ ($X \in \alpha$). </p> <p> 4. Аналогично, так как точка $Y$ лежит на прямой $b$ ($Y \in b$), а прямая $b$ лежит в плоскости $\alpha$ ($b \subset \alpha$), то и точка $Y$ лежит в плоскости $\alpha$ ($Y \in \alpha$). </p> <p> 5. Прямая $c$ проходит через точки $X$ и $Y$. Эти точки различны ($X \ne Y$), потому что если бы они совпадали, то эта общая точка принадлежала бы одновременно прямым $a$ и $b$, а значит, совпадала бы с точкой их пересечения $P$. Но по условию прямая $c$ не проходит через $P$. </p> <p> 6. Итак, мы имеем две различные точки прямой $c$ ($X$ и $Y$), которые лежат в плоскости $\alpha$. Согласно аксиоме стереометрии, если две точки прямой лежат в плоскости, то все точки этой прямой лежат в этой плоскости. Следовательно, прямая $c$ целиком лежит в плоскости $\alpha$ ($c \subset \alpha$). </p> <p> 7. Поскольку плоскость $\alpha$ была определена однозначно парой пересекающихся прямых $a$ и $b$, а прямая $c$ была выбрана произвольно из всех прямых, удовлетворяющих условию, то все такие прямые лежат в одной и той же плоскости $\alpha$. </p> <p> Таким образом, пропуски в тексте из задания заполняются следующим образом: </p> <p> «По <strong>теореме о двух пересекающихся прямых</strong> через пересекающиеся прямые $a$ и $b$ проходит некоторая плоскость $\alpha$, причем $X \in \alpha$ и $Y \in \alpha$, так как прямые $a$ и $b$ <strong>лежат в плоскости $\alpha$</strong>.» </p> <p> «Поэтому, согласно <strong>аксиоме о принадлежности прямой плоскости</strong>, прямая $XY$ лежит в плоскости $\alpha$. Итак, все рассматриваемые прямые лежат в <strong>одной и той же плоскости $\alpha$</strong>.» </p> <p> Ответ: Утверждение доказано. Все прямые, не проходящие через точку $P$ и пересекающие прямые $a$ и $b$, лежат в одной плоскости $\alpha$, которая однозначно определяется прямыми $a$ и $b$. </p>" ] #original: array:6 [ "id" => 1434269 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1085} "task" => array:2 [ "refs" => "738471" "type" => "task" ] "text" => "<p><strong>Доказательство.</strong></p> <p> Это утверждение является следствием основных аксиом и теорем стереометрии. Приведем развернутое доказательство. </p> <p> 1. Согласно теореме о существовании и единственности плоскости, проходящей через две пересекающиеся прямые, через прямые $a$ и $b$, которые пересекаются в точке $P$, можно провести единственную плоскость. Назовем эту плоскость $\alpha$. Таким образом, обе прямые, $a$ и $b$, целиком лежат в этой плоскости ($a \subset \alpha$ и $b \subset \alpha$). </p> <p> 2. Рассмотрим любую прямую (обозначим ее $c$), которая удовлетворяет условиям задачи: она не проходит через точку $P$, но пересекает прямую $a$ в некоторой точке $X$ и прямую $b$ в некоторой точке $Y$. </p> <p> 3. Так как точка $X$ лежит на прямой $a$ ($X \in a$), а прямая $a$ лежит в плоскости $\alpha$ ($a \subset \alpha$), то по определению принадлежности точка $X$ также лежит в плоскости $\alpha$ ($X \in \alpha$). </p> <p> 4. Аналогично, так как точка $Y$ лежит на прямой $b$ ($Y \in b$), а прямая $b$ лежит в плоскости $\alpha$ ($b \subset \alpha$), то и точка $Y$ лежит в плоскости $\alpha$ ($Y \in \alpha$). </p> <p> 5. Прямая $c$ проходит через точки $X$ и $Y$. Эти точки различны ($X \ne Y$), потому что если бы они совпадали, то эта общая точка принадлежала бы одновременно прямым $a$ и $b$, а значит, совпадала бы с точкой их пересечения $P$. Но по условию прямая $c$ не проходит через $P$. </p> <p> 6. Итак, мы имеем две различные точки прямой $c$ ($X$ и $Y$), которые лежат в плоскости $\alpha$. Согласно аксиоме стереометрии, если две точки прямой лежат в плоскости, то все точки этой прямой лежат в этой плоскости. Следовательно, прямая $c$ целиком лежит в плоскости $\alpha$ ($c \subset \alpha$). </p> <p> 7. Поскольку плоскость $\alpha$ была определена однозначно парой пересекающихся прямых $a$ и $b$, а прямая $c$ была выбрана произвольно из всех прямых, удовлетворяющих условию, то все такие прямые лежат в одной и той же плоскости $\alpha$. </p> <p> Таким образом, пропуски в тексте из задания заполняются следующим образом: </p> <p> «По <strong>теореме о двух пересекающихся прямых</strong> через пересекающиеся прямые $a$ и $b$ проходит некоторая плоскость $\alpha$, причем $X \in \alpha$ и $Y \in \alpha$, так как прямые $a$ и $b$ <strong>лежат в плоскости $\alpha$</strong>.» </p> <p> «Поэтому, согласно <strong>аксиоме о принадлежности прямой плоскости</strong>, прямая $XY$ лежит в плоскости $\alpha$. Итак, все рассматриваемые прямые лежат в <strong>одной и той же плоскости $\alpha$</strong>.» </p> <p> Ответ: Утверждение доказано. Все прямые, не проходящие через точку $P$ и пересекающие прямые $a$ и $b$, лежат в одной плоскости $\alpha$, которая однозначно определяется прямыми $a$ и $b$. </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" => "738472" "type" => "task" ] "previous" => array:2 [ "refs" => "738470" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1092 #items: array:1 [ 0 => App\Models\Book {#1030} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1203 #items: array:1 [ 0 => App\Models\BookPage {#1096 #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" => 971653 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "6" "field_url" => "/10-klass/geometrija/glazkov-tetrad/page-6" "field_display_title" => "6" "field_folder" => "1" "field_image_name" => "6" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1100 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1099 #items: array:1 [ 0 => App\Models\Book {#1101 #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" => 261 "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 {#1102 #items: array:1 [ 0 => App\Models\Term {#1103 #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 [ …10] #original: array:10 [ …10] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1104 #items: array:1 [ 0 => App\Models\Term {#1105 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1106 #items: array:1 [ 0 => App\Models\Term {#1107 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1108 #items: array:3 [ 0 => App\Models\Term {#1109 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false …18 } 1 => App\Models\Term {#1110 …30} 2 => App\Models\Term {#1111 …30} ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1112 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1113 #items: array:1 [ 0 => App\Models\Term {#1114 …30} ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1115 #items: array:1 [ 0 => App\Models\Term {#1116 …30} ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1117 #items: array:1 [ 0 => App\Models\Term {#1118 …30} ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1119 #items: array:1 [ 0 => App\Models\Term {#1120 …30} ] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1121 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1122 #items: array:1 [ 0 => App\Models\Term {#1123 …30} ] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1124 #items: array:1 [ 0 => App\Models\Term {#1125 …30} ] #escapeWhenCastingToString: false } "field_publication_number" => "4" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1126 #items: array:1 [ 0 => App\Models\Term {#1127 …30} ] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1128 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => "в сеточку" "field_publication_year" => "2020" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное пособие" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "110" "field_priority" => "14" "field_default_folder" => "/geometrija_10/glazkov-rt/" "field_isbn" => "978-5-09-097573-5" "field_cover" => array:1 [ 0 => "/media/geometrija_10/glazkov-rt/covers/cover1.webp?ts=1739295144" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/geometrija_10/glazkov-rt/covers/cover1.webp?ts=1739295144" "alt" => "" "width" => "1200" "height" => "1661" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1129 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1130 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1131 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/geometrija/glazkov-tetrad" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1132 #items: array:1 [ 0 => App\Models\Term {#1133 …30} ] #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" => 261 "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 {#1102} "field_class" => Illuminate\Database\Eloquent\Collection {#1104} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1106} "field_author" => Illuminate\Database\Eloquent\Collection {#1108} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1112} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1113} "field_country" => Illuminate\Database\Eloquent\Collection {#1115} "field_city" => Illuminate\Database\Eloquent\Collection {#1117} "field_series" => Illuminate\Database\Eloquent\Collection {#1119} "field_umk" => Illuminate\Database\Eloquent\Collection {#1121} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1122} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1124} "field_publication_number" => "4" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1126} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1128} "field_under_the_edition_degree" => null "field_cover_description" => "в сеточку" "field_publication_year" => "2020" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное пособие" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "110" "field_priority" => "14" "field_default_folder" => "/geometrija_10/glazkov-rt/" "field_isbn" => "978-5-09-097573-5" "field_cover" => array:1 [ 0 => "/media/geometrija_10/glazkov-rt/covers/cover1.webp?ts=1739295144" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/geometrija_10/glazkov-rt/covers/cover1.webp?ts=1739295144" "alt" => "" "width" => "1200" "height" => "1661" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1129} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1130} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1131} "field_url" => "/10-klass/geometrija/glazkov-tetrad" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1132} "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 } "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 {#1134 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1135 #items: array:1 [ 0 => App\Models\Element {#1136 #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" => 1147769 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1137 #items: array:1 [ 0 => App\Models\Edition {#1138 …30} ] #escapeWhenCastingToString: false } "book_page" => array:2 [ "refs" => "971653" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "6-1.jpg" "alt" => null "width" => "1945" "height" => 2794 "path" => "/media/geometrija_10/glazkov-rt/0-1/6-1.webp?ts=1745219733" ] ] ] #original: array:6 [ "id" => 1147769 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1137} "book_page" => array:2 [ "refs" => "971653" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "6-1.jpg" "alt" => null "width" => "1945" "height" => 2794 "path" => "/media/geometrija_10/glazkov-rt/0-1/6-1.webp?ts=1745219733" ] ] ] #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" => "971654" "type" => "book_page" ] "previous" => array:2 [ "refs" => "971652" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1144 #items: array:3 [ 0 => App\Models\Task {#1145 #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" => 738470 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "6" "field_page_end" => null "field_url" => "/10-klass/geometrija/glazkov-tetrad/00-3" "field_display_title" => "3" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1146 #items: array:1 [ 0 => App\Models\Term {#1147 …30} ] #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 {#1148 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1149 #items: array:1 [ 0 => App\Models\Branch {#1150 …30} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1154 #items: array:1 [ 0 => App\Models\Branch {#1150 …30} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1155 #items: array:3 [ 0 => App\Models\Element {#1156 …30} 1 => App\Models\Element {#1158 …30} 2 => App\Models\Element {#1166 …30} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "738471" "type" => "task" ] "previous" => array:2 [ "refs" => "738469" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1174 #items: array:1 [ 0 => App\Models\Book {#1101} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "971653" "type" => "book_page" ] ] #original: array:24 [ "id" => 738470 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "6" "field_page_end" => null "field_url" => "/10-klass/geometrija/glazkov-tetrad/00-3" "field_display_title" => "3" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1146} "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 {#1148} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1149} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1154} "content" => Illuminate\Database\Eloquent\Collection {#1155} "next" => array:2 [ "refs" => "738471" "type" => "task" ] "previous" => array:2 [ "refs" => "738469" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1174} "page" => array:2 [ "refs" => "971653" "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 => "*" ] } 1 => App\Models\Task {#1175 #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" => 738471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "6" "field_page_end" => null "field_url" => "/10-klass/geometrija/glazkov-tetrad/00-4" "field_display_title" => "4" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1176 #items: array:1 [ 0 => App\Models\Term {#1147 …30} ] #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 {#1177 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1178 #items: array:1 [ 0 => App\Models\Branch {#1150 …30} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1179 #items: array:1 [ 0 => App\Models\Branch {#1150 …30} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1180 #items: array:3 [ 0 => App\Models\Element {#1181 …30} 1 => App\Models\Element {#1183 …30} 2 => App\Models\Element {#1185 …30} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "738472" "type" => "task" ] "previous" => array:2 [ "refs" => "738470" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1187 #items: array:1 [ 0 => App\Models\Book {#1101} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "971653" "type" => "book_page" ] ] #original: array:24 [ "id" => 738471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "6" "field_page_end" => null "field_url" => "/10-klass/geometrija/glazkov-tetrad/00-4" "field_display_title" => "4" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1176} "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 {#1177} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1178} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1179} "content" => Illuminate\Database\Eloquent\Collection {#1180} "next" => array:2 [ "refs" => "738472" "type" => "task" ] "previous" => array:2 [ "refs" => "738470" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1187} "page" => array:2 [ "refs" => "971653" "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 => "*" ] } 2 => App\Models\Task {#1188 #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" => 738472 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "6" "field_page_end" => null "field_url" => "/10-klass/geometrija/glazkov-tetrad/00-5" "field_display_title" => "5" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1189 #items: array:1 [ 0 => App\Models\Term {#1147 …30} ] #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 {#1190 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1191 #items: array:1 [ 0 => App\Models\Branch {#1150 …30} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1192 #items: array:1 [ 0 => App\Models\Branch {#1150 …30} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1193 #items: array:3 [ 0 => App\Models\Element {#1194 …30} 1 => App\Models\Element {#1196 …30} 2 => App\Models\Element {#1198 …30} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "738473" "type" => "task" ] "previous" => array:2 [ "refs" => "738471" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1200 #items: array:1 [ 0 => App\Models\Book {#1101} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "971653" "type" => "book_page" ] ] #original: array:24 [ "id" => 738472 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "6" "field_page_end" => null "field_url" => "/10-klass/geometrija/glazkov-tetrad/00-5" "field_display_title" => "5" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1189} "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 {#1190} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1191} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1192} "content" => Illuminate\Database\Eloquent\Collection {#1193} "next" => array:2 [ "refs" => "738473" "type" => "task" ] "previous" => array:2 [ "refs" => "738471" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1200} "page" => array:2 [ "refs" => "971653" "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 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:21 [ "id" => 971653 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "6" "field_url" => "/10-klass/geometrija/glazkov-tetrad/page-6" "field_display_title" => "6" "field_folder" => "1" "field_image_name" => "6" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1100} "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1099} "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 {#1134} "content" => Illuminate\Database\Eloquent\Collection {#1135} "next" => array:2 [ "refs" => "971654" "type" => "book_page" ] "previous" => array:2 [ "refs" => "971652" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1144} ] #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" => 738471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "6" "field_page_end" => null "field_url" => "/10-klass/geometrija/glazkov-tetrad/00-4" "field_display_title" => "4" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#331} "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} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#342} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1066} "content" => Illuminate\Database\Eloquent\Collection {#1067} "next" => array:2 [ "refs" => "738472" "type" => "task" ] "previous" => array:2 [ "refs" => "738470" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1092} "page" => array:2 [ "refs" => "971653" "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 => "*" ] }
№4 (с. 6)
Условие. №4 (с. 6)
скриншот условия
На рисунке прямые $a$ и $b$ пересекаются в точке $P$. Докажите, что все прямые, не проходящие через точку $P$ и пересекающие прямые $a$ и $b$ в каких-то точках $X$ и $Y$, лежат в одной плоскости.
Доказательство.
По через пересекающиеся прямые $a$ и $b$ проходит некоторая плоскость $\alpha$, причем $X \in \alpha$ и $Y \in \alpha$, так как прямые $a$ и $b$.
Поэтому, согласно , прямая $XY$ лежит в плоскости $\alpha$. Итак, все рассматриваемые прямые лежат в .
Решение. №4 (с. 6)
Решение 2. №4 (с. 6)
Доказательство.
Это утверждение является следствием основных аксиом и теорем стереометрии. Приведем развернутое доказательство.
1. Согласно теореме о существовании и единственности плоскости, проходящей через две пересекающиеся прямые, через прямые $a$ и $b$, которые пересекаются в точке $P$, можно провести единственную плоскость. Назовем эту плоскость $\alpha$. Таким образом, обе прямые, $a$ и $b$, целиком лежат в этой плоскости ($a \subset \alpha$ и $b \subset \alpha$).
2. Рассмотрим любую прямую (обозначим ее $c$), которая удовлетворяет условиям задачи: она не проходит через точку $P$, но пересекает прямую $a$ в некоторой точке $X$ и прямую $b$ в некоторой точке $Y$.
3. Так как точка $X$ лежит на прямой $a$ ($X \in a$), а прямая $a$ лежит в плоскости $\alpha$ ($a \subset \alpha$), то по определению принадлежности точка $X$ также лежит в плоскости $\alpha$ ($X \in \alpha$).
4. Аналогично, так как точка $Y$ лежит на прямой $b$ ($Y \in b$), а прямая $b$ лежит в плоскости $\alpha$ ($b \subset \alpha$), то и точка $Y$ лежит в плоскости $\alpha$ ($Y \in \alpha$).
5. Прямая $c$ проходит через точки $X$ и $Y$. Эти точки различны ($X \ne Y$), потому что если бы они совпадали, то эта общая точка принадлежала бы одновременно прямым $a$ и $b$, а значит, совпадала бы с точкой их пересечения $P$. Но по условию прямая $c$ не проходит через $P$.
6. Итак, мы имеем две различные точки прямой $c$ ($X$ и $Y$), которые лежат в плоскости $\alpha$. Согласно аксиоме стереометрии, если две точки прямой лежат в плоскости, то все точки этой прямой лежат в этой плоскости. Следовательно, прямая $c$ целиком лежит в плоскости $\alpha$ ($c \subset \alpha$).
7. Поскольку плоскость $\alpha$ была определена однозначно парой пересекающихся прямых $a$ и $b$, а прямая $c$ была выбрана произвольно из всех прямых, удовлетворяющих условию, то все такие прямые лежат в одной и той же плоскости $\alpha$.
Таким образом, пропуски в тексте из задания заполняются следующим образом:
«По теореме о двух пересекающихся прямых через пересекающиеся прямые $a$ и $b$ проходит некоторая плоскость $\alpha$, причем $X \in \alpha$ и $Y \in \alpha$, так как прямые $a$ и $b$ лежат в плоскости $\alpha$.»
«Поэтому, согласно аксиоме о принадлежности прямой плоскости, прямая $XY$ лежит в плоскости $\alpha$. Итак, все рассматриваемые прямые лежат в одной и той же плоскости $\alpha$.»
Ответ: Утверждение доказано. Все прямые, не проходящие через точку $P$ и пересекающие прямые $a$ и $b$, лежат в одной плоскости $\alpha$, которая однозначно определяется прямыми $a$ и $b$.
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 класс, для упражнения номер 4 расположенного на странице 6 к рабочей тетради серии мгу - школе 2020 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 №4 (с. 6), авторов: Глазков (Юрий Александрович), Юдина (Ирина Игоревна), Бутузов (Валентин Фёдорович), ФГОС (старый) базовый и углублённый уровень обучения учебного пособия издательства Просвещение.