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 4 of 5: Pair the second two triangles
Detailed analysis
Let R be the circumradius and b=CA. Since CD=b cos C, angle OCD=90°−A, and OC=R, the area of OCD is (1/2)Rb cos C cos A. The same computation gives the identical area for OAF, because AF=b cos A and angle OAF=90°−C.