Circular geometry problem

Question

Let ABCD be a convex quadrilateral such that the diagonals AC and BD are perpendicular, and let P be their intersection. Prove that the reflections of P with respect to AB, BC, CD, DA are concyclic.

Answer 1

Let $P_{A}$ be the projection of $P$ onto $AB$, $P_B$ be the projection of $P$ onto $BC$, and so on. Note that $P_{A}P_{B}P_{C}P_{D}$ is an dilation of the quadrilateral we want to prove cyclic, so instead we shall prove this one cyclic.

Note that since $PP_ABP_B$ is a cyclic quadrilateral, we have $\angle PP_AP_B=\angle PBP_B=\angle P_BPP_C.$ Similarly, we have $\angle PP_AP_D=\angle DPP_D$, $\angle PP_CP_B=\angle BPP_B$, and $\angle PP_CP_D=\angle APP_D$. This means that

$\angle P_DP_AP_B+\angle P_BP_CP_D=\angle P_B P_B+\angle P_BP_C+\angle P_DP_A+\angle P_DP_D\\=\angle BP_C+\angle AP_D=90^\circ+90^\circ=180^\circ$ so $P_AP_BP_CP_D$ is cyclic as desired.