Question

Prove $\frac{1}{r_1}+\frac{1}{r_2}+\frac{1}{r_3}-\frac{1}{r}=0$

Answer 1

$\frac{1}{r_1}+\frac{1}{r_2}+\frac{1}{r_3}=\frac{1}{r}$

Since,

$\begin{array}{c}{r}_1=\frac{\Delta }{s-a}\\ r_2=\frac{\Delta }{s-b}\\ r_3=\frac{\Delta }{s-c}\\ r=\frac{\Delta }{s}\\ \frac{1}{r_1}+\frac{1}{r_2}+\frac{1}{r_3}\\ =\frac{s-a}{\Delta }+\frac{s-b}{\Delta }+\frac{s-c}{\Delta }\\ =\frac{3s-\left(a+b+b\right)}{\Delta }\\ =\frac{3s-2s}{\Delta }=\frac{s}{\Delta }=\frac{1}{r}\end{array}$