Moon over mountains

A rectangular frame encloses two congruent equilateral triangles and a unit circle. What is difference between width and height of the rectangle?

Scroll down for a solution to this problem.


The height is 2/√3 units larger than the width.

Suppose a is the side length of the triangles. Then draw three radii to the tangent lines as shown and two other line segments. Using the dimensions in the 30-60-90 triangles and the Tangent-tangent theorem, one can derive distances as shown.

Now the width equals a+1-1/√3, while the height is a+1+1/√3. Subtracting them cancels the a+1 and leaves 2/√3.


Belladonna was inspired by this picture:

La lune est retenue
Dans ses pics de montagnes enlacés
Elle rêve, de se lever, de quitter,
Ces monts où elle est conviée
Pour briller soirée après soirée
Mais c’est difficile
Mirangu l’a enfermée
Lui seul a la clé de sa cage azurée.


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