View source for 2007 IMO Shortlist Problems/G3

==IMO 2007 Short list/G3==
[[File:Trapezoid 17.png|500px|right]]
The diagonals of a trapezoid <math>ABCD</math> intersect at point <math>P.</math>

Point <math>Q</math> lies between the parallel lines <math>BC</math> and <math>AD</math> such that <math>\angle AQD = \angle CQB,</math> and line <math>CD</math> separates points <math>P</math> and <math>Q.</math>

Prove that <math>\angle BQP = \angle DAQ.</math>

<i><b>Proof</b></i>

<math>\angle AQD = \angle CQB \implies</math>

<math>BQ</math> and <math>AQ</math> are isogonals with respect <math>\angle CQD.</math>

<math>P =AC \cap BD, BC || AD \implies</math>

<math>QS || AD</math> is isogonal to <math>QP</math> with respect <math>\angle CQD.</math>

From the converse of <i><b>The isogonal theorem</b></i> we get

<math>\angle BQP = \angle SQA = \angle DAQ.</math>  <math>\blacksquare</math>
*[[Isogonal conjugate]]
'''vladimir.shelomovskii@gmail.com, vvsss'''