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


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Explanation

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Copyright © 1996-2017 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.

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

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

Orthodiagonal Quadrilaterals

  1. Invariance in Orthodiagonal Quadrilaterals
  2. Orthodiagonal and Cyclic Quadrilaterals
  3. Classification of Quadrilaterals
  4. Pythagorean Theorem in an Orthodiagonal Quadrilateral
  5. Easy Construction of Bicentric Quadrilateral
  6. Easy Construction of Bicentric Quadrilateral II

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Copyright © 1996-2017 Alexander Bogomolny

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