# A Construction Comes in Handy

One of my friends shared a geometry problem with me, already quite despondent after having tried it for long. The problem statement was :

In the square $ABCD$ below,  $\angle EAF = 45^{\circ}$. Prove that $EF^2 = DE^2 + FB^2$.

Starting off…

Unsurprisingly, I spent the first twenty minutes applying trigonometry and classical geometry to solve the problem. However, when the size of the calculations became unreasonable and it seemed like that I was going in circles, I decided to change tactics. The problem statement is so reminiscent of the Pythagorean theorem, that I decided to look and if necessary construct a right angle triangle.The word “construct” here is key. Once that is done… well! The enthusiastic reader should spare a spirited attack at the problem with this hint (which may not be very helpful) before reading on.

The problem is virtually solved after the following construction:

Construction comes in handy!

After this it is a matter of filling in the details. The seasoned problem-solver may already be grinning widely.

Once you have the solution it seems almost laughable! Well that is almost always the case  when one sees an elegant solution using just simple and beautiful ideas.