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Giải bài 6 trang 93 sách bài tập toán 9 - Chân trời sáng tạo tập 1

Cho tam giác nhọn ABC có (widehat {BAC} = {45^o}) và có các đỉnh nằm trên đường tròn (O). Các đường cao BH, CK cắt đường tròn (O) tại D, E. Chứng minh ba điểm D, O, E thẳng hàng.

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 \(\widehat {BAC} = {45^o}\) và có các đỉnh nằm trên đường tròn (O). Các đường cao BH, CK cắt đường tròn (O) tại D, E. Chứng minh ba điểm D, O, E thẳng hàng.

Phương pháp giải - Xem chi tiết

Dựa vào các góc nội tiếp cùng chắn một cung thì bằng nhau để chứng minh DE là đường kính.

Lời giải chi tiết

Ta có \(BH \bot AC\) nên tam giác ABH vuông tại H. Mà \(\widehat {BAH} = {45^o}\) nên \(\widehat {ABH} = {45^o}\). Mặt khác \(\widehat {ABD} = \widehat {ACD}\) (góc nội tiếp cùng chắn \(\overset\frown{AD}\)) nên \(\widehat {ACD} = {45^o}\)   (1) Ta có \(CK \bot AB\) nên tam giác ACK vuông tại K. Mà \(\widehat {CAK} = {45^o}\) nên \(\widehat {ACK} = {45^o}\)   (2) Từ (1) và (2) suy ra \(\widehat {DCE} = {90^o}\) nên DE là đường kính. Vậy ba điểm D, O, E thẳng hàng.

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

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