Let ABC be a triangle and D the foot of the altitude from A. Let E and F lie on a line passing through D such that AE is perpendicular to BE, AF is perpendicular to CF, and E and F are different from D. Let M and N be the midpoints of the segments BC and EF, respectively. Prove that AN is perpendicular to NM.
(In reply to
re: Possible approach (Spoiler) by Harry)
Maybe it is too much of a leap, Harry.
As far as I can reconstruct from my diagram, I actually started with the 3 point circle on A,M,N first, then checked that AM was its diameter (which seemed pretty likely given that D is on the same circle and ADM is a right angle). I thought that was probably enough, but if it isn't I'm quite happy to concede the point!
Edited on December 27, 2012, 3:36 pm

Posted by broll
on 20121227 15:28:25 