MathNet 2024-part-01 — 168 problems (EN–VI) Question 72

Let $ABC$ be an acute triangle with midpoint $M$ of $AB$. The point $D$ lies on the segment $MB$ and $I_1, I_2$ denote the incenters of $\triangle ADC$ and $\triangle BDC$. Given that $\angle I_1MI_2 = 90^\circ$, show that $CA = CB$.