Let $ABC$ be a right angled triangle with $\angle A = 90^o$and $BC = a$, $AC = b$, $AB = c$. Let $d$ be a line passing trough the incenter of triangle and intersecting the sides $AB$ and $AC$ in $P$ and $Q$, respectively. (a) Prove that $$b \cdot \left( \frac{PB}{PA}\right)+ c \cdot \left( \frac{QC}{QA}\right) =a$$(b) Find the minimum of $$\left( \frac{PB}{PA}\right)^ 2+\left( \frac{QC}{QA}\right)^ 2$$
Problem
Source: 2010 Saudi Arabia BMO TST 5.3 - Balkan
Tags: geometry, ratio, geometric inequality, right triangle