Prove that in every acute triangle, the equation stands: $h_c = c \cfrac{\tan(\alpha)\cdot \tan(\beta)}{\tan(\alpha)+\tan(\beta)} $

Question

Prove that in every acute triangle, the equation stands: $$h_c = c \cfrac{\tan(\alpha)\cdot \tan(\beta)}{\tan(\alpha)+\tan(\beta)} $$

Answer 1

Given

$$ \begin{eqnarray} \tan(\alpha) &=& \frac{h_c}{c_1},\\\\ \tan(\beta) &=& \frac{h_c}{c_2},\\\\ c &=& c_1 + c_2. \end{eqnarray} $$

Then

$$ \begin{eqnarray} c &=& \frac{h_c}{\tan(\alpha)} + \frac{h_2}{\tan(\beta)}\\\\ &=& h_c \frac{\tan(\alpha)+\tan(\beta)}{\tan(\alpha)\tan(\beta)} \end{eqnarray} $$

Whence

$$ h_c = c \frac{\tan(\alpha)\tan(\beta)}{\tan(\alpha)+\tan(\beta)} $$