I used to think algebraic geometry and analytic geometry are the same thing. How naive was I?
Given \(\triangle ABC\), let \(BE,CF\) be angle bisectors such that \(E\in AC, F\in AB\). Reflect the incenter \(I\)of \(ABC\) over \(BC\) to obtain \(I'\). \(G,H\in BC\) such that \(I'G\perp BE, I'H\perp CF\). Prove that \(\angle FHB=\angle EGC\)