Notes
two triangles in two circles solution

Solution to the Two Triangles in Two Circles Problem

Solution using Cyclic Quadrilaterals

Consider the following diagram, in which two extra lines have been drawn.

The extra lines create two cyclic quadrilaterals, which mean that angles $E\hat{D}C$ and $C\hat{B}E$ are supplementary angles (that is, they add up to $180^\circ$). Similarly, angles $A\hat{F}E$ and $E\hat{B}A$ also add up to $180^\circ$. Angles $C\hat{B}E$ and $E\hat{B}A$ are angles on a straight line which make up a half turn and so add up to $180^\circ$.

We therefore have that $E\hat{D}C$ and $A\hat{F}E$ are supplementary angles, and so the line segments $AF$ and $CD$ are parallel as $E\hat{D}C$ and $A\hat{F}E$ are cointerior angles.

This means that the quadrilateral $ACDF$ is a trapezium and so the areas of the triangles $AOC$ and $DOF$ are the same since they are the side triangles in a crossed trapezium.