Apollonius’ Theorem

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.

20130807 Triangle Def 0

In this post, I derive the following relationship, called Apollonius’ Theorem:

20130807 eqApollonius 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).

20130807 Triangle Def 1 20130807 Triangle Def 2

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.:

20130807 Three Triangs

For T1:

20130807 eq1

For T2:

20130807 eq2

For T3:

20130807 eq3Let [eq.1] = [eq.2]:

20130807 eq4

Let [eq.2] = [eq.3]:

20130807 eq5

Let [eq.4] = [eq.5]:

20130807 eqQED

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2 Responses to Apollonius’ Theorem

  1. navalator says:

    I don’t get one bit of this. How can i use this information in my everyday life, especially when comparing prices in supermarkets?

    • cjkfung says:

      I don’t think this theorem can be applied in the everyday lives of laymen. Outside of mathematics, this theorem might be useful in architecture or computer programming.

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