A rectangle inside a square. One of its sides is tangent to the quarter circle. What fraction is green as x approaches 0?

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## Solution

The fraction in the limit x↓0 is 1/2.

Recognize the three tangency points: upper left and lower right square vertex and point E. From the tangent-tangent theorem we can infer the distances x and y as shown.

Using the Pythagorean theorem in the upper right triangle, we get that y=(1-x)/(1+x), leading to x+y=(1+x^{2})/(1+x). Using similarity of this triangle and the one in the lower right corner we get z/x=(x+y)/(1-y), leading to z=(1+x^{2})/2.

Taking the limit x↓0, the long rectangle side tends to 1 and the short one tends to 1/2, leading to a fraction of 1/2.

## Poem

A green bloc balances slowly

In a big square closed,

So lonely

The bloc is not free

He wants liberty

And leave this figures so unselfishly

It’s green colour like a flower

Makes him nice and full of power

Who will help and really care?

Maybe you or me, or someone there .