Question 20
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
$$
\frac{EI}{FI} \cdot \frac{ED}{FD} = \frac{EI_1^2}{FI_2^2}.
$$
$$
\frac{EI}{FI} \cdot \frac{ED}{FD} = \frac{EI_1^2}{FI_2^2}.
$$