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 \(s^2 >= w * h\).
For all \(w, h\): \[\begin{aligned} s^2 - w * h &= p^2/4 - w*h \\ &= \frac{w^2 + 2wh + h^2}{4} - w*h \\ &= \frac{w^2 - 2wh + h^2}{4} \\ &= \frac{(w - h)^2}{4} \geq 0 \end{aligned}\]
We have shown that \(s^2 >= w * h\), 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.