When I was studying calculus, all the geometry problems that asked for the maximum area of a $k$-polygon given a fixed perimeter, or the minimum perimeter of a $k$-polygon given a fixed area, would always have the regular $k$-polygon as the solution. After a lot of exercises, I noticed that this always held true, so I assumed that it was a general fact and used it as a shortcut for giving the correct answer. However, I was not able to prove or disprove this assumption, so I want to ask here for a proof.
Although this result was not mentioned in my book, I still think it is true and had been discovered centuries ago, so I want to ask who was the first person to have ever discovered this and proven it. (Since this problem is elementary I think it is probably Newton or someone who came before Newton.)