Two congruent concentric equilateral triangles. Prove that the gold fraction is red/blue.

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## Solution

Because the triangles are congruent and concentric, symmetry implies that all six non-overlap triangles are congruent. They have sides a, b and c and an angle of 60° between sides a and c. Their area is √3ac/4.

It is easy to show that the total area is that of one equilateral plus three of those smaller triangles, whereas the gold overlap region is one equilateral minus three of these triangles.

So, total area T=√3(x^{2}+3ac)/4 where x=a+b+c is the side length of the equilateral triangles. The overlap region O=√3(x^{2}-3ac)/4. Working out the fraction, we get O/T=b/(a+c). But red is 3b and blue is 3(a+c). This completes the proof.

## Poem

The christmas star

A hope from above

It’s like a sun

Shining and guiding us

The bride of the moon

She is the messager

Of a more fraternal and freer world

In peace

No more war

It is a symbol of sharing

And of solidarity actions

Tonight it lights up

The proud Xmas tree