ELMO 2023 Problem 3

Convex quadrilaterals $ABCD$, $A_1B_1C_1D_1$, and $A_2B_2C_2D_2$ are similar with vertices in order. Points $A$, $A_1$, $B_2$, $B$ are collinear in order, points $B$, $B_1$, $C_2$, $C$ are collinear in order, points $C$, $C_1$, $D_2$, $D$ are collinear in order, and points $D$, $D_1$, $A_2$, $A$ are collinear in order. Diagonals $AC$ and $BD$ intersect at $P$, diagonals $A_1C_1$ and $B_1D_1$ intersect at $P_1$, and diagonals $A_2C_2$ and $B_2D_2$ intersect at $P_2$. Prove that points $P$, $P_1$, and $P_2$ are collinear.

Proposed by Holden Mui