A quarter circle containing two touching semicircles cut by a chord. What fraction is brown?

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## Solution

A fraction of 13/36*(1-2/pi) is brown, which is approximately 0,13.

The first part of the solution is to find the radii of the semicircles. Setting the radius of the quarter circle to 1, the larger semicircle has 1/2. From the right triangle in the centre and the Pythagorean theorem we get for the smaller one x=1/3.

The next step is to recognize that the brown circular segments are similar to each other and to the one of the quarter circle spanning 90°. This is due to the fact that the three circles touch each other and the chord is passing through the tangency points.

Now a 90° circular segment has area r^{2}(π-2)/4. Thus, we get for the brown area (1/4+1/9)*(π-2)/4. Dividing this by the total area π/4 we arrive at the required fraction.

## Poem

Geometrical poet Belladonna associated freely on The dark side:

Le soleil est éclipsé

Par la lune

Le coeur est affamé

Vidé le sourire

Aucune peur de mourir

Dans Starwars et sa guerre

On sabote

L’étoile au super laser

Planète stellaire détruite

Aussitôt reconstruite

Dans la lune forestière

Bella