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Giải bài tập 7.12 trang 38 SGK Toán 9 tập 2 - Cùng khám phá

Cho tam giác nhọn ABC có AD, BE, CF là đường cao và H là trực tâm. Chứng minh rằng a) Tứ giác AEHF, BDHF và CDHE là các tứ giác nội tiếp b) DA là đường phân giác của góc FDE.

Tổng hợp Đề thi vào 10 có đáp án và lời giải

Toán - Văn - Anh
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Đề bài

☂Cho tam giác nhọn ABC có AD, BE, CF là đường cao và H là trực tâm. Chứng minh rằng

✃a) Tứ giác AEHF, BDHF và CDHE là các tứ giác nội tiếp

b) DA là đường phân giác của góc FDE.

Phương pháp giải - Xem chi tiết

Đọc kĩ dữ kiện để vẽ hình Trong một tứ giác nội tiếp, tổng số đo hai góc đối nhau bằng \({180^o}\). Áp dụng hai góc nội tiếp cùng chắn một cung thì bằng nhau.

Lời giải chi tiết

💝a) Ta có \(\widehat {AEH} = \widehat {AFH} = {90^o}\) (Do CF và BE là đường cao)

suy ra AEHF là tứ giác nội tiếp.

🦂Chứng minh tương tự BDHF và CDHE là các tứ giác nội tiếp

🌳b) Theo phần a ta có BDHF nội tiếp nên \(\widehat {ABE} = \widehat {FDA}\)

𒊎DHEC nội tiếp nên \(\widehat {ADE} = \widehat {FCA}\).

𝄹Lại có \(\widehat {ABE} = \widehat {FCA}\) (cùng phụ \(\widehat {BAC}\))

💟Suy ra \(\widehat {FDA} = \widehat {ADE}\) hay AD là đường phân giác của góc FDE.

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Tham Gia Group Dành Cho Lớp 9 - Ôn Thi Vào Lớp 10 Miễn Phí

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