A circle and a right triangle. What’s the angle α?

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## Solution

The angle α is 45°.

First, draw the line segment EF. From Thales we know it is a diagonal of the circle. From the midpoint theorem we also know it is parallel to DC and half its length. This enables us to basically fit three copies of triangle EFB on top of it as shown.

Now consider radius AG. Because DC is tangent, angle AGD is right. From the parallelness we infer also EAG is right. Therefore EFG is isosceles and since its top angle is 90°, the other two angles must be 45°. Triangle DGE is congruent, so this makes alpha is 45° as well.

## Poem

I’m so small

And so kind

I like when you play with me

And go for some walks

I’m glad to stay with you

As for me you do

My heart is unselfish

And has only one wish

Just know you’re there

And for me you care

Still after your caresses longing

And for your love searching