BMO 2010 Shortlist

Let $A{H}_a$ be an altitude in the triangle $ABC$ and $P_a$ be its midpoint. Define analogously $H_b$, $P_b$, etc.

It is easy to see that the point $A^{\prime }$ divides the segment $P_b{P}_c$ in the same ratio that $A_1$ divides $BC$. By Menelaus' theorem for the triangle $P_a{P}_b{P}_c$, claim (a) follows.

Consider an affine transformation mapping of the triangle $ABC$ onto the triangle $P_a{P}_b{P}_c$. When $l$ contains a fixed point $X$, $l^{\prime }$ contains the fixed point $Y$ whose affine coordinates with respect to triangle $P_a{P}_b{P}_c$ equal the affine coordinates of $X$ with respect to $ABC$. We are now left to show that $X\equiv O\iff Y\equiv O_9$.

This is easiest to do by considering two special cases, say, when $l$ contains some vertex of the triangle $ABC$.

Another approach is this: Let $Z$ be the point whose affine coordinates with respect to triangle $H_a{H}_b{H}_c$ equal the affine coordinates of $X$ with respect to triangle $ABC$. Clearly, $Y$ is the midpoint of $XZ$. Let $O^{\prime }$ be the circumcenter of triangle $H_a{H}_b{H}_c$. It is clear that $A{O}^{\prime }H{H}_a$ is a straight line and by similar figures $A{H}_b{H}_c{O}^{\prime }\sim ABCO$ we see that $AO$ divides $BC$ and $H_{a}H$ divides $H_b{H}_c$ in equal ratios. It follows that $X=O\iff Z=H$. $\square$