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 5 of 5: Pair the remaining triangles
Detailed analysis
Cyclically exchanging A, B, and C in the preceding calculation gives [OCE]=[OBF]. Thus all six regions form the three required equal-area pairs.