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
$\frac{1}{P_1}+\frac{1}{P_2}+\frac{1}{P_3}=\frac{1}{r}$

Answer 1

$\begin{array}{c}Areaoftriangle,\Delta =\frac{1}{2}AP\\ \frac{1}{P_1}=\frac{a}{2\Delta }\\ similarly\\ \frac{1}{P_2}=\frac{b}{2\Delta }and\frac{1}{p_3}=\frac{c}{2\Delta }\\ areaof\Delta intermofinradius\\ \Delta =r\cdot s,\\ where,25=a+b+c\\ \frac{1}{P_1}+\frac{1}{P_2}+\frac{1}{P_3}=\frac{a+b+c}{2\Delta }=\frac{25}{2\times r\cdot s}=\frac{1}{r}\end{array}$

Hence, proved.