# Apollonius's Theorem: Statement and Proof of Apollonius'Theorem

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In mathematics, theorems are the statements which have proven results based on the previously set statements, like theorems and generally confirmed statements like axioms. The theorems are known as the results which are proven to be accurate from the set of different axioms. This term is used where these axioms are of numerical logic with the systems in the form of the question.

## Statement and Proof of Apollonius’ Theorem

Medians are known to form the most important sets of the components in the geometry of triangles which are closely related to the triangle being independent of the geometric shapes.

In Apollonius’ Theorem, the relation between the medians and the sides of the triangle are known. Apollonius’ theorem is a kind of theorem which relates to the length of the median of a triangle to the length of their sides.

### Apollonius’ Theorem Statement

Statement- *“*the sum of squares of any of the two sides of a triangle equals to twice its square on half of the third side, along with the twice of its square on the median bisecting the third side*”*

Or

If O is the midpoint of one of the sides of the triangle (LMN), then prove that

### Apollonius’ Theorem Proof

Choose the origin of the rectangular form of the Cartesian coordinates at the point O and the axis coming along the sides and also as y – axis. If in case , then the coordinates of the points M, as well as N, are and respectively. If coordinates of the point L are (b, c), then

, (Since the coordinates of the point O are )

also,

Therefore,

Example:

In . Find the length of the median from A to BC.

Solution:

As given, let the median be AD.

We can apply Apollonius Theorem to say that

Here given,

Here we put the value in equation, (1)

Hence, the value of median AD is .

Example:

In , and Median from A to BC is 14, Find the value of AC.

Solution:

As given, let the median be AD.

We can apply Apollonius Theorem to say that

Here given,

Here we put the value in equation, (1)

Hence the value of AC is