A circle inside a quarter circle. The triangle connects the two tangency points and the top of the quarter circle. What’s the angle α?

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## Solution

The angle α is 45°.

This fact uses the fact that lines from the common tangency point have a chord that span the same angle in both circles. The lines from the small circle can therefore be extended to the bottom and leftmost point of the large one.

Now using the inscribed angle theorem, we can infer the 45° angle on the right. Using Thales, the total must be 90°, so our wanted angle is also 45°.

## Poem

An aluminium hat

That protects from the sun’s rays

And from cold and rain anyway

Is used until today

A great white shiny metal

To handle so easy

For tasks used so ordinary

In the past also in jewellery

Is a metal made for poetry

Only known by a minority

Is a rag bag likely