AIME I 2009 #5

[stupidityismygam]

Triangle [unparseable or potentially dangerous latex formula] has [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] Points [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] are located on [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] respectively so that [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] is the angle bisector of angle [unparseable or potentially dangerous latex formula] Let [unparseable or potentially dangerous latex formula] be the point of intersection of [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] and let [unparseable or potentially dangerous latex formula] be the point on line [unparseable or potentially dangerous latex formula] for which [unparseable or potentially dangerous latex formula] is the midpoint of [unparseable or potentially dangerous latex formula] If [unparseable or potentially dangerous latex formula] find [unparseable or potentially dangerous latex formula]

[stupidityismygam]

This will be hard without a diagram, but here we go!

Draw [unparseable or potentially dangerous latex formula] Note that [unparseable or potentially dangerous latex formula], so by SAS [unparseable or potentially dangerous latex formula]

So, we have that [unparseable or potentially dangerous latex formula]. So, by AAA, [unparseable or potentially dangerous latex formula]

This means that [unparseable or potentially dangerous latex formula]

By the angle bisector theorem, [unparseable or potentially dangerous latex formula] This means that [unparseable or potentially dangerous latex formula]

So, [unparseable or potentially dangerous latex formula]