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 3 of 5: Use the two midpoint theorems
Detailed analysis
The rectangle QBCR has M as the midpoint of BC, and MP is parallel to QB. Hence P is the midpoint of QR. Since N is the midpoint of EF, the segment joining the midpoints of the diagonals EF and QR in trapezoid QEFR is parallel to its bases; therefore NP is parallel to QE (and to RF).