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 1 of 5: Build three auxiliary rectangles
Detailed analysis
Choose P so that ADMP is a rectangle. On line AP choose Q and R so that QBDA and ADCR are rectangles. Since Q, B, D lie on the circle with diameter AB, and R, C, D lie on the circle with diameter AC, the perpendicular hypotheses give that ADEQ and ADFR are cyclic.