# Triangle in Circle in Triangle in Semi-Circle +-- {.image} [[TriangleinCircleinTriangleinSemiCircle.png:pic]] > A triangle in a circle in a triangle in a semicircle ... what's the angle? =-- ## Solution by [[Angle in a Semi-Circle]] and [[Alternate Segment Theorem]] +-- {.image} [[TriangleinCircleinTriangleinSemiCircleLabelled.png:pic]] =-- With the points labelled as above, angle $A \hat{E} C$ is the [[angle in a semi-circle]] so is a [[right-angle]]. As $A E$ and $C E$ are both tangent to the smaller circle, $F E$ and $E D$ are the same length, so triangle $F E D$ is [[isosceles]] and [[right-angled triangle|right-angled]]. Therefore, angle $D \hat{F} E$ is $45^\circ$. By the [[alternate segment theorem]], angle $D \hat{B} F$ is also $45^\circ$.