1. Introduction Previously, I had written a post detailing the Fermat point of a triangle: the unique point which minimizes the sum of distances to the triangle’s vertices. Recently, I realized a small oversight I had made in my proof of the case where all of the triangle’s angles are less than . In today’s post, we will be patching this oversight, so it is recommended to read through the original post before continuing (as a challenge, try to find…
Introduction Given any triangle , suppose we take each side and and construct an equilateral triangle on it. You may notice that given a line segment, there are two ways to construct an equilateral triangle on it (one on each side); we will eventually consider both ways, but for now suppose each triangle is constructed on the outside of , i.e. they do not overlap with . We now consider finding the centers of each of these equilateral triangles, as…
Introduction This post is a basic derivation of the famous Heron’s formula, which allows calculation of the area of any triangle in terms of its side lengths . We refer to the following diagram: As is convention, let sides be opposite to angles . Further, let the altitude length and . Then, by the Pythagorean Theorem applied to triangles and , we have the following two equations: Subtracting the two equations cancels the and yields ….