Problem 1
Let ABC be an acute triangle with altitudes AD, BE, and CF, and let O be the center of its circumcircle. Show that the segments OA, OF, OB, OD, OC, OE dissect triangle ABC into three pairs of triangles having equal areas.
Step 3 of 5: Pair the first two triangles
Detailed analysis
Combining the two ratios and cancelling OA, OC, and AB as in the official solution gives BD·OM=AE·ON. Since OBD and OAE have bases BD and AE and corresponding heights OM and ON, their areas are equal.