Question 180
Let $ABC$ be a right angled triangle with $\widehat{A}=90^{\circ}$ and $BC=a$, $AC=b$, $AB=c$. Let $d$ be a line passing through the incenter of triangle and intersecting the sides $AB$ and $AC$ in $P$ and $Q$, respectively.
a. Prove that
$$
b \cdot \frac{PB}{PA} + c \cdot \frac{QC}{QA} = a ;
$$
b. Find the minimum of
$$
\left(\frac{PB}{PA}\right)^{2} + \left(\frac{QC}{QA}\right)^{2} .
$$
a. Prove that
$$
b \cdot \frac{PB}{PA} + c \cdot \frac{QC}{QA} = a ;
$$
b. Find the minimum of
$$
\left(\frac{PB}{PA}\right)^{2} + \left(\frac{QC}{QA}\right)^{2} .
$$