# Solution to the [[Two Quarter Circles]] Puzzle +-- {.image} [[TwoQuarterCircles.jpeg:pic]] > Two quarter circles. What’s the angle? =-- ## Solution by [[Angle at the Circumference is Half the Angle at the Centre]] and [[Angles in the Same Segment]] +-- {.image} [[TwoQuarterCirclesLabelled.jpeg:pic]] =-- With the points labelled as above, the blue circle has centre $G$ and passes through $C$, $D$, and $E$. The point $F$ will end up lying on this circle, but this needs to be shown. The reflex angle $A \hat{B} C$ is $270^\circ$, so since the [[angle at the circumference is half the angle at the centre]], angle $A \hat{F} C$ is $135^\circ$. Then from [[angles at a point on a straight line]], angle $C \hat{F} D$ is $45^\circ$. This is the same as angle $C \hat{E} D$ and so from the converse to [[angles in the same segment]] being equal, $F$ lies on the same circle as $E$, $C$, and $D$. Finally, since [[angles in the same segment]] are equal, angle $D \hat{F} E$ is the same as angle $D \hat{C} E$, which is $45^\circ$. Hence angle $D \hat{F} E$ is $45^\circ$.