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Orthodiagonal and Cyclic Quadrilaterals: What Is This About?
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Copyright © 1996-2008 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. |
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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.
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In triangles APD and BPC we have
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which indeed shows that quadrilateral XYZW is cyclic.
References
- T. Andreescu, R. Gelca, Mathematical Olympiad Challenges, Birkhäuser, 2004, 5th printing, 1.2.5 (p. 9)
Copyright © 1996-2008 Alexander Bogomolny
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