A Concurrency from Six Pedal Points
Let ABC be a triangle and P and Q be two other points. Drop perpendiculars from P to the three sides of ABC. Let X, Y, and Z be the feet of the perpendiculars (pedals) of Q on the three perpendiculars. Let X', Y', and Z' be the pedals of P on AQ, BQ, and CQ. Then XX', YY", and ZZ' are concurrent.
The theorem is contained in "The Theorem on the Six Pedals", available on Darij Grinberg's home page.