Question

If $P_1,P_2,P_3$ are the altitudes drawn from vertices $A,B,C$ to the opposite sides of a triangle respectively, then show that
$P_1{P}_2{P}_3=\frac{(abc{)}^2}{8R^3}=\frac{8{\Delta }^3}{abc}$

Answer 1

$\begin{array}{c}Areaof\Delta =\frac{1}{2}AP\\ P_1=\frac{2\Delta }{a}\\ P_2=\frac{2\Delta }{b}\\ P_3=\frac{2\Delta }{c}\\ \therefore P_1{P}_2{P}_3=\frac{2\Delta }{a}\cdot \frac{2\Delta }{b}\cdot \frac{2\Delta }{c}=\frac{8{\Delta }^3}{abc}\end{array}$

Hence, proved.