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 1 of 5: Introduce the side midpoints

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M=\operatorname{mid}(BC),\quad N=\operatorname{mid}(AC),\quad \angle MOC=\angle EAB,\quad \angle OMC=\angle AEB=90^\circ.

Detailed analysis

Let M and N be the midpoints of BC and AC. Since O is the circumcenter, OM is perpendicular to BC; the central and inscribed angle relations give angle MOC=angle EAB, while both OMC and AEB are right angles.