Problem A. 630. (December 2014)
A. 630. The konvex quadrilateral \(\displaystyle ABCD\) has an inscribed circle with center \(\displaystyle I\). The rays \(\displaystyle AB\) and \(\displaystyle DC\) meet at point \(\displaystyle F\), the rays \(\displaystyle AD\) and \(\displaystyle BC\) meet at point \(\displaystyle G\). Let \(\displaystyle \mathcal{E}\) be the ellipse with foci \(\displaystyle F\) and \(\displaystyle G\) that passes through points \(\displaystyle B\) and \(\displaystyle D\), and let \(\displaystyle \mathcal{H}\) be the hyperbola branch with foci \(\displaystyle F\) and \(\displaystyle G\) that passes through points \(\displaystyle A\) and \(\displaystyle C\). Denote by \(\displaystyle P\) and \(\displaystyle Q\) the intersections of \(\displaystyle \mathcal{E}\) and \(\displaystyle \mathcal{H}\). Show that the points \(\displaystyle P\), \(\displaystyle Q\) and \(\displaystyle I\) are collinear.
(5 pont)
Deadline expired on January 12, 2015.
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