Two congruent circles and a right triangle inside a semicircle. What fraction is blue?

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## Solution

The blue fraction is 256/625.

We set the radius of the two circles to 1. We can use the Pythagorean theorem in the right triangle to find that a+b-ab+1=0. Next, we use the fact that the triangle FIG is isosceles. A proof of this interesting lemma is given below.

This gives us a+b=a+2, so b=2. From the previous equation it follows that a=3, so the right triangle is a 3-4-5 triangle.

Using similarity of the two smaller triangles easily gives c=5/4. The radius of the semicircle r=a/2+c/2+1=25/8. The requested fraction follows.

## Poems

The birdwatcher

Knows many thrilling sights

He hunts pictures patiently

As I do for words in poetry

A poet and protector naturally

Just peace to work,

No hurry

Waiting patiently

He wins ultimately

Loving birds and words

Is a world full of mystery.