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Cho tam giác nhọn MNP. Các trung tuyến ME và NF cắt nhau tại G. Trên tia đối của tia FN lấy điểm D sao cho FN = FD.

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Đề bài

Cho tam giác nhọn MNP. Các trung tuyến ME và NF cắt nhau tại G. Trên tia đối của tia FN lấy điểm D sao cho FN = FD. a) Chứng minh rằng \(\Delta \)MFN = \(\Delta \)PFD b) Trên đoạn thẳng FD lấy điểm H sao cho F là trung điểm của GH. Gọi K là trung điểm của GK. Chứng minh rằng ba điểm M, H, K thẳng hàng.

Video hướng dẫn giải

Phương pháp giải - Xem chi tiết

a) Chứng minh \(\Delta \)MFN = \(\Delta \)PFD theo trường họp cạnh góc cạnh Sử dụng tính chất của điểm đối xứng qua một điểm, trung điểm của 1 đoạn thẳng và 2 góc đối đỉnh b) Chứng minh H là trọng tâm của tam giác MPD sau đó dựa vào tính chất ta suy ra M, H, K thẳng hàng

Lời giải chi tiết

a) Vì N đối xứng với D qua F (theo giả thiết)Nên NF = DF (1)Vì F là trung điểm của MP (theo giả thiết)Nên MF = PF (2)Vì góc NFM và góc PFD ở vị trí đối đỉnh nên 2 góc bằng nhau (3)Từ (1), (2) và (3) \( \Rightarrow \)\(\Delta \)MFN = \(\Delta \)PFD (c-g-c)b) Xét tam giác MPD có :F là trung điểm MD,K là trung điểm DP (theo giả thiết)Mà 2 đường trung tuyến của tam giác MPD là DF và MK cắt nhau tại H\( \Rightarrow \) H là trọng tâm \(\Delta \)MPD\( \Rightarrow \) M, H, K thẳng hàng

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