# Solution to the Two Triangles in a Circle and Semi-Circle Puzzle +-- {.image} [[TwoTrianglesinaCircleandSemiCircle.png:pic]] > Which of the shaded triangles has the largest area? =-- ## Solution by [[Cyclic quadrilateral]], [[Angles at a point on a straight line]], and [[Crossed trapezium]] +-- {.image} [[TwoTrianglesinaCircleandSemiCircleLabelled.png:pic]] =-- In the above diagram, quadrilateral $A B C F$ is [[cyclic quadrilateral|cyclic]] (in the semi-circle) and so angles $A \hat{F} C$ and $C \hat{B} A$ add up to $180^\circ$ (that is, are [[supplementary]]). For the same reason, angles $C \hat{F} E$ and $E \hat{D} C$ also add up to $180^\circ$. Since [[angles at a point on a straight line]] add up to $180^\circ$, angles $A \hat{F} C$ and $C \hat{F} E$ add up to $180^\circ$. Putting all those together, angles $E \hat{D} C$ and $C \hat{B} A$ add up to $180^\circ$. This establishes $E D$ and $A B$ as [[parallel]] and so quadrilateral $A B D E$ is a [[trapezium]]. As detailed at [[crossed trapezium]], the two triangles therefore have the same area.