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Giải bài 68 trang 85 sách bài tập toán 8 – Cánh diều

Cho tam giác \(ABC\) có ba góc nhọn, điểm \(I\) thuộc cạnh \(BC\) và \(IM,IN\) lần lượt là đường phân giác của các góc \(AIC\) và \(AIB\). Chứng minh: \(AN.BI.CM=BN.IC.AM\).

ꦦTổng hợp đề thi học kì 2 lớp 8 tất cả các môn - Cánh diều

Toán - Văn - Anh - Khoa học tự nhiên
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Đề bài

Cho tam giác \(ABC\) có ba góc nhọn, điểm \(I\) thuộc cạnh \(BC\) và \(IM,IN\) lần lượt là đường phân giác của các góc \(AIC\) và \(AIB\). Chứng minh: \(AN.BI.CM=BN.IC.AM\).

Phương pháp giải - Xem chi tiết

Tính chất đường phân giác của tam giác: trong tam giác, đường phân giác của một góc chia cạnh đối diện thành hai đoạn thẳng tỉ lệ với hai cạnh kề hai đoạn ấy.

Lời giải chi tiết

Áp dụng tính chất đường phân giác vào các tam giác \(ABI,AIC\) ta có: \(\frac{AN}{NB}=\frac{AI}{BI};\frac{CM}{MA}=\frac{IC}{AI}\).Suy ra \(\frac{BI}{IC}.\frac{AN}{NB}.\frac{CM}{MA}=\frac{BI}{IC}.\frac{AI}{BI}.\frac{IC}{AI}=1\)Do đó \(AN.BI.CM=BN.IC.AM\).

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