KöMaL 2023 B5351

Let $P$ be an arbitrary point inside the equilateral triangle $ABC$. The line parallel to $AB$ through $P$ intersects sides $BC$ and $AC$ in points $C_1$ and $C_2$, respectively. Similarly, the parallel through point $P$ with side $BC$ intersects sides $AC$ and $AB$ in points $A_1$ and $A_2$, respectively. Finally, let the parallel through point $P$ with side $AC$ intersect sides $AB$ and $BC$ in points $B_1$ and $B_2$, respectively. Prove that the areas of triangles $A_1B_1C_1$ and $A_2B_2C_2$ are equal.