Let $ ABC$ be a triangle and $P$ a random point on the same plane as the triangle. Let $l$ be a line passing through $P$. Let $A_1,B_1,C_1$ be the intersection points of $BC,CA,AB$ with the reflections of $AP,BP,CP$ across line $l$ in respective order. Prove that $A_1,B_1,C_1$ are collinear.
First reflect $\Delta ABC$ across $l$. Suppose the image is $A'B'C'$. Now notice that these two triangles are perspective from the point at infinity. Hence the corresponding sides intersect at points that are collinear. These collinear points lie on $l$. Furthermore, let $PA',PB',PC'$ intersect $BC,CA,AB$ at $A_1,B_1,C_1$ respectively. We have to show that these three points are collinear. After that, I have tried to look for more perspective triangles but haven't been able to find anything that helps me.
How do I proceed? Am I overthinking?
Thanks in advance.