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 5 of 5: Conclude the perpendicularity
Detailed analysis
Since ADNP and ADMN are cyclic, angles ANP and ADM are supplementary in the first cyclic quadrilateral, and the parallel construction identifies the required angle at N. In the cyclic quadrilateral ADMN, the angle ANM equals 180 degrees minus ADM. Because AD is perpendicular to DM, ADM is 90 degrees, so ANM is 90 degrees and AN is perpendicular to NM.