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Giải bài 46 trang 92 SBT toán 10 - Cánh diều

Cho tam giác nhọn ABC có các cạnh đôi một khác nhau. Gọi H, O lần lượt là trực tâm và tâm đường tròn ngoại tiếp của tam giác, D là điểm đối xứng với H qua O. Chứng minh \(\overrightarrow {HA} + \overrightarrow {HB} + \overrightarrow {HC} = \overrightarrow {HD} \)

Tổng h💫ợp đề thi học kì 2 lớp 10 tất cả các môn - Cánh diều

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Đề bài

Cho tam giác nhọn ABC có các cạnh đôi một khác nhau. Gọi H, O lần lượt là trực tâm và tâm đường tròn ngoại tiếp của tam giác, D là điểm đối xứng với H qua O. Chứng minh \(\overrightarrow {HA}  + \overrightarrow {HB}  🍰+ \overrightarrow {HC}  = \overrightarrow {HD} \)

Phương pháp giải - Xem chi tiết

Bước 1: Lấy E đối xứng với A qua O

Bước 2: Chứng minh các tứ giác ADEH, BHCE là hình bình hành

Bước 3: Áp dụng quy tắc hình bình hành để chứng minh \(\overrightarrow {HA}  + \overrightarrow {HB}  + \overrightarrow {HC}  = \overrightarrow {HD} \)

Lời giải chi tiết

Gọi E là điểm đối xứng với A qua O . Khi đó AE là đường kính của đường tròn ngoại tiếp ∆ABC

Tứ giác ADEH có O là trung điểm HD và AE nên là hình bình hành

\( \Rightarrow \overrightarrow {HA}  + \overrightarrow {HE}  = \overrightarrow {HD} \)(1)Lại có: \(\widehat {ACE}\) là góc nội tiếp chắn nửa đường tròn nên \(\widehat {ACE} = {90^0}\)\( \Rightarrow EC \bot AC\), mà \(BH \bot AC\)\( \Rightarrow EC//BH\)Chứng minh tương tự ta có \(BE//HC\)Tứ giác BHCE có \(EC//BH\), \(BE//HC\) nên là hình bình hành\( \Rightarrow \overrightarrow {HB}  + \overrightarrow {HC}  = \overrightarrow {HE} \)(2)Từ (1) và (2) suy ra \(\overrightarrow {HA}  + \overrightarrow {HB}  + \overrightarrow {HC}  = \overrightarrow {HA}  + \overrightarrow {HE}  = \overrightarrow {HD} \) (ĐPCM) 

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