Three squares and two line segments. What’s the angle α?
Scroll down for a solution to this problem.
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
