Question

The vertical poles of heights $h_1,h_2$ and $h_3$ at the vertices A,B and C of a $△ABC$ subtend angles$\alpha ,\beta$and $\gamma$ respectively at the circumcentre of the triangle,If $\cot \alpha ,\cot \beta$ and $\cot \gamma$ are in AP then $h_1,h_2,h_3$ are in

• A. AP
• B. GP
• C. HP
• D. none of these.

Answer 1

Answer: (C)

HP

$\frac{R}{h_1}=\cot \alpha$ , etc where R is circumradius

As $\cot \alpha ,\cot \beta ,\cot \gamma$ is A.P

$\Rightarrow \frac{1}{h_1},\frac{1}{h_2},\frac{1}{h_3}$ are in A.P