# Cherchez le quadrilatère cyclique

What is this about?

A Mathematical Droodle

What if applet does not run? |

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Copyright © 1996-2018 Alexander BogomolnyThe applet is supposed to illustrate the following problem:

Let ABCD be a cyclic quadrilateral. At A and D construct equal angles BAE and CDF in directions opposite relative to BC. Let E and F lie on BC. Then quadrilateral AEFD is cyclic,

In a slightly different form the problem appeared in [Andreescu & Gelca, p. 9, problem 1.2.7]:

Points B and C are given on the side BC of a convex quadrilateral AEFD (with B closer to E than to F.) It is known that

The conclusion is valid even when points B and C coalesce whereas line BC is replaced by the tangent to the circle. With this in mind, the problem appears to generalize the one of an accidental angle bisector. Also, in this formulation, the requirement of convexity is quite spurious.

What if applet does not run? |

Suffice it to show that ∠ADF + ∠AEF = 180°. This is so because

= (∠ADC + ∠CDF) + (∠ABF - ∠BAE) | |

= (∠ADC + ∠ABC) + (∠CDF - ∠BAE) | |

= ∠ADC + ∠ABC | |

= 180°. |

### References

- T. Andreescu, R. Gelca,
*Mathematical Olympiad Challenges*, Birkhäuser, 2004.

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Copyright © 1996-2018 Alexander Bogomolny