A square inscribed in a triangle. Two areas are given. What is the square’s area?

Scroll down for a solution to this problem.

## Solution

The square area is 2√xy.

This problem can be solved in an algebraic way, as was shown by Dr Rick. The diagram contains three similar triangles: the unshaded ones and the big one. Their hypotenuses have proportions √x : √y : √x+√y. Their areas have proportions as the squares of that and since (√x+√y)^{2}=x+y+2√x√y, the square must have area 2√xy.

Note that this proof also holds in case the square would be parallelogram.

## Poem

The hammock

Keeps our dreams

Keeps our thoughts

From morning

To evening

We lay softly dreaming

Eyes closed, and shivering

Our hands sliding

Our bodies slowly travelling

A bird is kindly singing

A moment of eternity

Far from reality

The hammock is friendly

## One reply on “The hammock”

2 (xy)^.5