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

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

Bella