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BdMO Online Forum • View topic - IGO 2016 Advanced/4

For discussing Olympiad level Geometry Problems

4. In a convex quadrilateral \$ABCD\$, the lines \$AB\$ and \$CD\$ meet at point \$E\$ and the lines \$AD\$ and \$BC\$ meet at point \$F\$. Let \$P\$ be the intersection point of diagonals \$AC\$ and \$BD\$. Suppose that \$w_1\$ is a circle passing through \$D\$ and tangent to \$AC\$ at \$P\$. Also suppose that \$w_2\$ is a circle passing through \$C\$ and tangent to \$BD\$ at \$P\$. Let \$X\$ be the intersection point of \$w_1\$ and \$AD\$, and \$Y\$ be the intersection point of \$w_2\$ and \$BC\$. Suppose that the circles \$w_1\$ and \$w_2\$ intersect each other in \$Q\$ for the second time. Prove that the perpendicular from \$P\$ to the line \$EF\$ passes through the circumcenter of triangle \$XQY\$ .
I think we judge talent wrong. What do we see as talent? I think I have made the same mistake myself. We judge talent by the trophies on their showcases, the flamboyance the supremacy. We don't see things like determination, courage, discipline, temperament.

Thamim Zahin

Posts: 98
Joined: Wed Aug 03, 2016 5:42 pm