# Circles Resting on Triangles +-- {.image} [[CirclesRestingonTriangles.png:pic]] > Two semicircles are balanced on these three identical triangles. What's the missing area? =-- ## Solution by Lengths in an [[Isosceles]] [[Right-Angled Triangle]], [[Angles in a Triangle]], and [[Angles at a Point on a Straight Line]] +-- {.image} [[CirclesRestingonTrianglesLabelled.png:pic]] =-- The triangles are identical and $A B C D$ is a straight line, so angles $F \hat{B} A$ and $C \hat{B} F$ are the same and add to $180^\circ$, hence both are [[right-angles]]. Then angles $F \hat{C} B$ and $D \hat{C} E$ are equal to the non-right-angles, so since the [[angles in a triangle]] add up to $180^\circ$, these must add up to $90^\circ$. This leaves $90^\circ$ for angle $E \hat{C} F$. Since $F C$ and $E C$ are the same length, triangle $F C E$ is therefore an [[isosceles]] [[right-angled triangle]]. The length of $F E$ is then $\sqrt{2}$ times that of $F C$ and so the length [[scale factor]] from the dark blue semi-circle to the cyan one is $\sqrt{2}$. The [[area scale factor]] is thus $2$ and so the area of the cyan semi-circle is $28$.