Three semicircles with diameters a, b and x. Express x in terms of a and b.

Solution

Diameter x=a+b.

It can be shown that the line segment connecting the centres of the smaller semicircles is parallel to the diameter of the large semicircle. Now the length can be filled in as above. Clearly, if the large centre is halfway the triangle base and x=a+b, we have a solution. Moving it to the left or to the right, one cannot satisfy the conditions, so it must be unique.

Poem

My better half
An union for love and laugh
How can I know that the other half is you ?
This question is fair
What can we do
What can we share ?
Many sides one cannot see
And some sides to hide, I agree
But don’t worry, you are in my head
Sometimes in blue and Sometimes in red

Bella