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 2 of 5: Obtain two similarities

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\triangle OMC\sim\triangle AEB,\quad \frac{OM}{AE}=\frac{OC}{AB};\qquad \triangle ONA\sim\triangle BDA,\quad \frac{ON}{BD}=\frac{OA}{BA}.

Detailed analysis

The right-angle pairs and one equal acute angle make OMC similar to AEB. The analogous comparison of ONA and BDA gives the second similarity and its ratio.