A right triangle with an inscribed square. What is the area of the triangular part on the right?

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## Solution

The area x is 6.

First we deduce a general theorem by Thanos Kalogerakis relating areas x, y en z. These triangles are all similar. Name the common smallest angle α and set the square side to 1. The right-angled sides can be found as shown.

Now x=1/(2tan(α)), y=tan(α)/2 and z=sin(α)cos(α)/2. Calculate 1/x+1/y=2tan(α)+2/tan(α)=2sin(α)/cos(α)+2cos(α)/sin(α)=2/(sin(α)cos(α))=1/z. So we have the equation 1/x+1/y=1/z. Using y=3 and z=2, it follows that x=6.

## Visual solution

## Poem

The awning tent

Stretching tightly and transparent

Used to cover thermal solar

And sometimes also in winter

Aluminium is the structure

Against cold and warm factors

Awning is like a canopy

A woven mat like a velarium

As deployed above Colosseum.

Protecting tent and aluminium