Problem 4
Let ABC be a triangle and let D be the foot of the altitude from A. Let E and F be points on a line 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 BC and EF, respectively. Prove that AN is perpendicular to NM.
Step 4 of 5: Find two cyclic quadrilaterals
Detailed analysis
The parallel-side relations just obtained make ADNP a cyclic quadrilateral, with its sides parallel to the corresponding sides of ADFR. The rectangle ADMP has all four vertices on its circumcircle; using the same parallel relations shows that A, D, M, N are also concyclic.