Problem

Source: Turkish TST 2011 Problem 4

Tags: geometry, incenter, circumcircle, ratio, geometry proposed



Let $D$ be a point different from the vertices on the side $BC$ of a triangle $ABC.$ Let $I, \: I_1$ and $I_2$ be the incenters of the triangles $ABC, \: ABD$ and $ADC,$ respectively. Let $E$ be the second intersection point of the circumcircles of the triangles $AI_1I$ and $ADI_2,$ and $F$ be the second intersection point of the circumcircles of the triangles $AII_2$ and $AI_1D.$ Prove that if $AI_1=AI_2,$ then \[ \frac{EI}{FI} \cdot \frac{ED}{FD}=\frac{{EI_1}^2}{{FI_1}^2}.\]