Question 99
In triangle $ABC$, let $A'$ be the point symmetric to $A$ with respect to the circumcenter $O$ of $ABC$.
a) Prove that the sum of the squares of the tangent segments drawn from $A$ and $A'$ to the incircle of $ABC$ is equal to $4R^2 - 4Rr - 2r^2$.
b) Let $I$ be the incenter of $\triangle ABC$. Prove that the circle with center $A'$ and radius $A'I$ meets the circumcircle of $ABC$ at a point $L$ such that $AL = \sqrt{AB \cdot AC}$.
a) Prove that the sum of the squares of the tangent segments drawn from $A$ and $A'$ to the incircle of $ABC$ is equal to $4R^2 - 4Rr - 2r^2$.
b) Let $I$ be the incenter of $\triangle ABC$. Prove that the circle with center $A'$ and radius $A'I$ meets the circumcircle of $ABC$ at a point $L$ such that $AL = \sqrt{AB \cdot AC}$.