# Orthodiagonal and Cyclic Quadrilaterals

What Is This About?

A Mathematical Droodle

What if applet does not run? |

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Copyright © 1996-2018 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.

What if applet does not run? |

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 | |

= 90° = 90° | |

= 180°, |

which indeed shows that quadrilateral XYZW is cyclic.

### References

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

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Copyright © 1996-2018 Alexander Bogomolny