Câu hỏi của Anh Khương Vũ Phương

Question

Cho tam giác nhọn ABC, 3 đường cao AH, BI, CK

CM: S_HIK= (1 - cos²A - cos²B - cos²C).S_ABC

Trần Minh Hoàng · Accepted Answer

Ta có:

$\Delta AIK\sim\Delta ABC\left(g.g\right)\Rightarrow\frac{S_{AIK}}{S_{ABC}}=\left(\frac{AI}{AB}\right)^2=c\text{os}^2A$.

Tương tự: $\frac{S_{BHK}}{S_{ABC}}=c\text{os}^2B;\frac{S_{CIH}}{S_{ABC}}=c\text{os}^2C$.

Do đó: $\frac{S_{HIK}}{S_{ABC}}=1-c\text{os}^2A-c\text{os}^2B-c\text{os}^2C\Rightarrow...\Rightarrow\text{đ}pcm$

Câu trả lời:

Ta có:

$\Delta AIK\sim\Delta ABC\left(g.g\right)\Rightarrow\frac{S_{AIK}}{S_{ABC}}=\left(\frac{AI}{AB}\right)^2=c\text{os}^2A$.

Tương tự: $\frac{S_{BHK}}{S_{ABC}}=c\text{os}^2B;\frac{S_{CIH}}{S_{ABC}}=c\text{os}^2C$.

Do đó: $\frac{S_{HIK}}{S_{ABC}}=1-c\text{os}^2A-c\text{os}^2B-c\text{os}^2C\Rightarrow...\Rightarrow\text{đ}pcm$