Question

In ΔABC, Show that a = (r₂ + r₃) $\sqrt{\frac{r.r_1}{r_2.r_3}}$.

Answer 1

We take

$RHS=(r_{2}+r_{3})\sqrt{\frac{r.r_{1} }{r_{2}.r_{3}}}\\\\RHS=(\frac{\Delta}{s-b} +\frac{\Delta}{s-c})\sqrt{\frac{(s-b)(s-c)}{s.(s-a)}$

$RHS=\Delta(\frac{1}{s-b} +\frac{1}{s-c})\sqrt{\frac{(s-b)(s-c)(s-b)(s-c)}{s.(s-a)(s-b)(s-c)}$

$RHS=\Delta(\frac{s-c+s-b}{(s-b)(s-c)}).\frac{(s-b)(s-c)}\Delta}\\  \\RHS=s-c+s-b\\\\RHS=s+s-b-c\\\\RHS=2s-b-c\\\\RHS=a+b+c-b-b\\\\RHS=a=LHS$