Pedal quadrilateral and related topics

Abstract The paper studies properties of a pedal quadrilateral of an arbitrary point P with respect to a given quadrilateral ABCD . A specific formula for the area of the pedal quadrilateral is given, followed by conditions under which the pedal quadrilateral of a point P can be a parallelogram, a rectangle, or a square. Finally, certain properties are discussed when a circle can be circumscribed about quadrilateral ABCD . Specifically, a proof is provided that reflection line passes through the intersection of diagonals and through the anticenter of the cyclic quadrilateral. Proofs of all statements are synthetic, unlike most of proofs in the cited works, which were performed using an analytical method.