A circle with three tangents and two squares. The accentuated vertex is a tangency point. What’s the angle α?

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## Solution

The angle α is 90°.

We begin by setting the circle radius to 1 for convenience. Then using an angle chase, the distance BF=tan(22,5)=√2-1. This makes the smaller square of size √2.

Using the cosine rule in triangle CFE then leads to CF=√3. Thus the larger square has diagonal √6. Now the Pythagorean Theorem in right triangle EFH leads to FH=2.

Now tan(β)=1/(√2+1)=√2-1 , leading to β=22,5°. A simple angle chase then leads to angle α=90°.

## Poem

Kindergarten

The first year of school

For children aged five

A place where you can stay

A place where you can go

And where you can show what you know

Every day

The child must stay

To help him learn

And help him know

To cherish him and embrace him

And believe in his talent