An interesting relationship found in geometry involves the measurements of the sides of a triangle and the measurement of the triangle’s median. In the following diagram the median of the triangle (with sides X and Y) is indicated by Z. The median bisects the opposite side yielding measurements of a1 = a2 = a.
In this post, I derive the following relationship, called Apollonius’ Theorem:
Actually, this post was inspired by an article from a blog I’ve been following. I wanted to confirm the relationship myself (less formally and maybe more concisely?). With this theorem, we can calculate the length of a triangle’s median given the measurements of all the sides of any triangle.
First, we need to define the height h, and length b, both of which will help us when applying the Pythagorean Theorem. We can use either definition below and derive the same result amongst x, y, a and z (Apollonius’ theorem).
The following is working that can be used to derive the theorem based on the definitions in the triangle immediately above this paragraph. Let the left triangle be T1, the middle triangle T2 and the right-most triangle be T3. i.e.:
Let [eq.2] = [eq.3]:
Let [eq.4] = [eq.5]: