Let $\Delta ABC$ be a scalene triangle with center $I$ of its inscribed circle. Points $A_1$,$B_1$, and $C_1$ are the points of tangency of the same circle with $BC$,$CA$, and $AB$ respectively. Prove that the circumscribed circles of $\Delta AIA_1$,$\Delta BIB_1$, and $\Delta CIC_1$ intersect in a common point, different from $I$.
Problem
Source: VIII International Festival of Young Mathematicians Sozopol 2017, Theme for 10-12 grade
Tags: geometry