Nemotron-IMO-Bench 2026 G27

Let $ABCDE$ be a convex pentagon and let $A_1,\ldots, E_1$ be the intersections of the diagonals $BD$ and $CE$, $CE$ and $DA$, $DA$ and $EB$, $EB$ and $AC$, $AC$ and $BD$. If four of the quadrilaterals $AB_1A_1B, BC_1B_1C, CD_1C_1D, DE_1D_1E, EA_1E_1A$ are cyclic, prove that the fifth one is cyclic as well.