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 2 of 5: Obtain a trapezoid
Detailed analysis
The cyclic quadrilaterals ADEQ and ADFR share side AD, while their other two sides lie on the same supporting lines DE and DF. Equal angles subtended by the common chord AD therefore give EQ parallel to RF. Thus Q, E, F, R form a trapezoid.