A triangle with side lengths a and b and an inscribed rhombus. What is the blue fraction in terms of a and b?

Scroll down for a solution to this problem.

## Solution

The blue fraction is 2ab/(a+b)^{2}.

The top triangles are similar to the whole, and are in proportion a:b, so their area (as a fraction of the whole) is A = (a²+b²)/(a+b)², so blue area is 1-A = 2ab/(a+b)².

## Poem

The inner diamond

The brightness of a diamond

Cannot be seen

Shining from inside

You must search it to find

No precious stone can materialize

To be a real good human being

Inner beauty shines from the heart

It is not a merchandise

And has

As much value as a rubie