Back when I went to middle school, math books loved to throw a problem at us where we had to maximize the area of a pasture with a given length of string.
It was interesting to solve the first time around, but after encountering it more and more, I came to think that this exercise was insipid and non-inspired. At some point, I wondered how nice it would be to have some theorem that would make it possible to solve this problem in a few seconds, bypassing the painstaking calculations otherwise required.
Since I found no theorem that fit the bill, and I was yet to prove one, I enlisted the help of my dad and we got on our way.
The gist of the problem is the following:
Fix R a rectangle of width w and height h.
Then, its perimeter P is equal to 2w + 2h and its area A is equal to w ∗ h. We fix p = w + h the half perimeter of the rectangle.
If the rectangle is a square, we know p/2 = s the side length of the square.
Then, to prove our theorem, it is enough to show that s² ≥ w ∗ h.
For all w, h:
s² - w ∗ h
= p²/4 - wh
= (w² + 2wh + h²)/4 - wh
= (w² - 2wh + h²)/4
= (w - h)²/4
≥ 0
We have shown that s² ≥ w ∗ h, it follows that the theorem enounced above is true.
Very proud of my achievement, I rewrote the proof on a clean sheet of paper and took it to class the next morning.
At the beginning of class, I showed it to my teacher and he accepted to check my proof. However, I did not get to use the theorem since we changed chapters soon after.