Three triangles are glued together. The green ones are based on a common line. What’s the angle α?

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## Solution

The angle α is either 60° or 72°.

Start by acknowledging that sides AC and ED are parallel. One can therefore construct the parallelogram ACDF^{1}Note that it is actually a rhombus.. Now draw the circle centred in A going through B, C, F and E. Using the Inscribed angle theorem, we have ∠CAF is twice ∠CEF. So α=2(180-2α), ergo α=72°.

It can, however, happen that AE is parallel to CD. The Inscribed angle theorem does not work anymore, since E and F coincide. In that case it is easily seen that all three triangles are equilateral and hence α=60°.

## Poem

Two brotherly pines

Two green pines

Lived side by side

In beautiful brotherhood

Until the day

When the greatest

Was chosen

To decorate

The Xmas market

The little one cried and moaned

He refused

And wrapped their roots together

Thus

There were two trees

At the christmas market