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