# Problem 2 from the IMO 2007, Second solution

Here is Problem 2 from the IMO 2007:

Consider five points A, B, C, D and E such that ABCD is a parallelogram and BCED is a cyclic quadrilateral. Let *i* be a line passing through A. Suppose that *l* intersects the interior of the segment DC at F and intersects line BC at G. Suppose also that *l* is the bisector of angle DAB.

The solution below is by Vo Duc Dien and is a significant simplification of the solution by Anton Batominovski.

Based on Simson-Wallace's theorem the feet of projections of E down to the three sides of triangle BCD (denoted M, L and K) are colinear as seen on the diagram.

Since L and K are the midpoints of FC and CG, respectively, LK||FG, and in triangle ACG, LK intersects AC at its midpoint M. Since ABCD is a parallelogram, M is also the midpoint of DB.

Therefore, there is only one unique point M to satisfy conditions that M is on DB and also collinear with K and L, and M is also the foot of E to DB.

Extend EM to cut the circle at I. Since EI is perpendicular to DB and

It follows that ∠ECI = 90°. Further,

(*) | ∠4 = ∠5. | |

In triangle EFL: | ∠1 + ∠2 + ∠3 = 90° | |

In triangle EFG: | 2(∠1 + ∠2 + ∠5) = 180° | |

or | ∠1 + ∠2 + ∠5 = 90°. | |

Therefore | ∠3 = ∠5. |

From (*), ∠3 = ∠4. In other words,

|Activities| |Contact| |Front page| |Contents| |Geometry|

Copyright © 1996-2018 Alexander Bogomolny63414671 |