Orthodiagonal and Cyclic Quadrilaterals: What Is This About?
A Mathematical Droodle

Explanation

Copyright © 1996-2010 Alexander Bogomolny

The applet below provides an illustration to a problem from an outstanding collection by T. Andreescu and R. Gelca:

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, and DA are concyclic.

The quadrilateral in question is a dilation with coefficient 2 of the quadrilateral formed by projections of P on the sides of quadrilateral ABCD. It suffices to prove that the latter is cyclic. Let X, Y, Z, W be the feet of perpendiculars from P to the sides AB, BC, CD, DA. The quadrilaterals AXPW, BYPX, CZPY, DWPZ are cyclic as having a pair of opposite right angles. From this we obtain the following identities:

WAP = WXP, PXY = PBY, YZP = YCP, PZW = PDW.

In triangles APD and BPC we have

WXY + WZY = WXP + PXY + YZP + PZW   = WAP + PDW + PBY + YCP   = 90o = 90o   = 180o,

which indeed shows that quadrilateral XYZW is cyclic.

References

1. T. Andreescu, R. Gelca, Mathematical Olympiad Challenges, Birkhäuser, 2004, 5^th printing, 1.2.5 (p. 9)

Copyright © 1996-2010 Alexander Bogomolny