A circle containing two touching semicircles. What is the proportion of the circle diameter a to the line segment b?

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## Solution

The proportion a : b is √2 : 1.

First, draw the line segments AB and AC. The first is the diameter of the larger semicircle, whereas the latter goes through the touchpoint. Therefore, they have an angle of 45°.

Now use the Inscribed angle theorem to infer that the angel BOC at the circle centre is 90°. Then it is simply a matter applying the Pythagorean theorem to find that b=a/√2.

## Poem

This problem inspired Belladonna to the following poem:

Le gobelet de feu

Pour certains

Pour d’autres un calice

Ouvert pour recevoir

Les abeilles qui butinent

Où le vase sacré du ciboire

Un creuset qui accueille

Les biens terrestres

Les formes généreuses

Une coupe heureuse

Belladonna