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Giải bài 2 trang 111 vở thực hành Toán 9 tập 2

Cho tam giác ABC nội tiếp đường tròn (O). Gọi M, N, P lần lượt là trung điểm của các cạnh BC, CA, AB. Chứng minh rằng các tứ giác ANOP, BPOM, CMON là các tứ giác nội tiếp.

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

Toán - Văn - Anh
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Đề bài

Cho tam giác ABC nội tiếp đường tròn (O). Gọi M, N, P lần lượt là trung điểm của các cạnh BC, CA, AB. Chứng minh rằng các tứ giác ANOP, BPOM, CMON là các tứ giác nội tiếp

Phương pháp giải - Xem chi tiết

+ Chứng minh OP, ON, OM lần lượt là các đường cao của các tam giác AOB, AOC, BOC. + Tứ giác ANOP có \(\widehat {ANO} = \widehat {APO} = {90^0}\) nên tứ giác ANOP nội tiếp đường tròn có tâm là trung điểm của AO và bán kính bằng \(\frac{{AO}}{2}\). + Chứng minh tương tự ta có BPOM, CMON cũng là các tứ giác nội tiếp.

Lời giải chi tiết

Do các tam giác AOB, AOC, BOC đều cân tại O nên OP, ON, OM lần lượt là các đường cao của các tam giác này. Do vậy, tứ giác ANOP có \(\widehat {ANO} = \widehat {APO} = {90^0}\). Do vậy tứ giác ANOP nội tiếp đường tròn có tâm là trung điểm của AO và bán kính bằng \(\frac{{AO}}{2}\). Tương tự BPOM, CMON cũng là các tứ giác nội tiếp.

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

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