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.

Screen Shot 2017-04-02 at 10.33.21 PM

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:

Screen Shot 2017-04-02 at 11.34.58 PM.png

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Construction comes in handy!

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

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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.


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