# 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. [[!include two triangles in two circles solution SVG]] 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]].