Problem

Source: ELMO 2023/3

Tags: Elmo, geometry



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