Circle and intersecting lines

[ins-]

Let $P$ is a random point chosen internally for the triangle $ABC$. $A_0$ is the intersection point of $AP$ with $BC$. $B_0$ is the intersection point of $BP$ with $AC$. $C_0$ is the intersection point of $CP$ with $AB$. Circumcircle of the triangle $A_{0}B_{0}C_{0}$ intersects the sides $AB$, $BC$, $CA$ respectively at the points $A_1$, $B_1$, $C_1$. Prove that $AA_1$, $BB_1$, $CC_1$ intersects at a common point.

[Sergic Primazon]

$\frac{AB_1}{AC_1}\frac{AB_0}{AC_0}=1$ , $\frac{BC_1}{BA_1}\frac{BC_0}{BA_0}=1$ ,$\frac{CA_1}{CB_1}\frac{CA_0}{CB_0}=1$

по т. Чевы : $\frac{AB_0BC_0CA_0}{AC_0BA_0CB_0}=1 \Rightarrow \frac{AB_1BC_1CA_1}{AC_1BA_1CB_1}=1 $