Orthodiagonal and Cyclic Quadrilaterals
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Copyright © 1996-2012 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
- T. Andreescu, R. Gelca, Mathematical Olympiad Challenges, Birkhäuser, 2004, 5th printing, 1.2.5 (p. 9)
Orthodiagonal Quadrilaterals
- Invariance in Orthodiagonal Quadrilaterals
- Orthodiagonal and Cyclic Quadrilaterals
- Classification of Quadrilaterals
- Pythagorean Theorem in an Orthodiagonal Quadrilateral
- Easy Construction of Bicentric Quadrilateral
- Easy Construction of Bicentric Quadrilateral II
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Copyright © 1996-2012 Alexander Bogomolny
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