Two quarter circles and a semicircle inside a circle. What fraction of the circle is shaded by the semicircle?

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## Solution

A fraction of 1/4 is shaded.

First draw line segments CA, BA and DA. From applying Thales in both semicircles we get two right angles. This implies that CD is a straight line.

From looking at angle DCB of 45° and applying the inscribed angle theorem, we find that angle DOB is right, where O is the circle centre.

From the inverse of Thales we get that O must also lie on the shaded semicircle. Now it is simply applying the Pythagoras theorem to learn that R^{2}=2r^{2}.

Please find here a nice Desmos calculator by Per Henrik Christiansen showing all possible configurations.

## Poem

The earth is blue like a orange

Whasps bloom green

A necklace of windows

Wings cover the leaves

You have all the joy of the sun

All the sun on the earth

On the way to your beauty

(translated from a poem from Paul Eluard)