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
