Question

Show that in the triangle ABC the altitudes $p_1,p_2,p_3$ are in AP then a, b, c are in HP.

Answer 1

Area of triangle $\left(\Delta \right)=$ $\frac{1}{2}$ base $\times$ height

$=\frac{1}{2}a{p}_1=\frac{1}{2}b{p}_2=\frac{1}{2}c{p}_3$

$p_1=\frac{2\Delta }{a},p_2=\frac{2\Delta }{b},p_3=\frac{2\Delta }{c}$

$\frac{2\Delta }{a},\frac{2\Delta }{b},\frac{2\Delta }{c}$ are in A.P.

$\therefore a,b,c$ are in H.P