Question 28
Let $I$ be the incenter of a triangle $ABC$. Let $F$ and $G$ be the feet of the perpendiculars drawn from $A$ to the lines $BI$ and $CI$, respectively. Rays $AF$ and $AG$ intersect the circumcircles of the triangles $CFI$ and $BGI$ second times at points $K$ and $L$, respectively. Prove that line $AI$ bisects the segment $KL$.
Source-figure note: the original illustration below belongs to an unrelated four-circle configuration. It is retained unchanged for source fidelity and is not used in the proof of this problem.
Figure 16
Source-figure note: the original illustration below belongs to an unrelated four-circle configuration. It is retained unchanged for source fidelity and is not used in the proof of this problem.
Figure 16