RS-SMO 2019 Problem 4

For a $\triangle ABC$ , let $A_1$ be the symmetric point of the intersection of angle bisector of $\angle BAC$ and $BC$ , where center of the symmetry is the midpoint of side $BC$ , In the same way we define $B_1$ ( on $AC$ ) and $C_1$ (on $AB$ ). Intersection of circumcircle of $\triangle A_1B_1C_1$ and line $AB$ is the set $\{Z,C_1 \}$ , with $BC$ is the set $\{X,A_1\}$ and with $CA$ is the set $\{Y,B_1\}$ . If the perpendicular lines from $X,Y,Z$ on $BC,CA$ and $AB$ , respectively are concurrent , prove that $\triangle ABC$ is isosceles.