Three squares and two line segments. What’s the angle α?

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## Solution

The angle α is 45°.

The argument rests on symmetry. First, we note that both inscribed squares are each others image under a point reflection in I followed by an appropriate scaling from that point. In particular, the square centres N and O are each others images. Thus I lies on the line segment connecting N and O.

Now from fourfold rotational symmetry we know that both N and O lie on the diagonal of the large square. Ergo, the angle α must be 45°.

## Poem

Two squares are secured

With bungee cordes

Tightly fastened

Like secured luggage

We can feel attached

By a bungee rope

And never loose hope

From earth to sky diving

Like bungee jumping

Dreaming and safely falling

On the ground

There is always a friend

Holding the string