Apollonius Theorem is an important theorem in *Geometry*. It connects the* lengths of the sides* of a triangle, to the *lengths of the medians*.

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## What is Apollonius Theorem?

The **median** of a triangle is the line segment connecting a **vertex** of a triangle to the *midpoint of the opposite side.* Apollonius Theorem connects the length of the median to the lengths of the bisected side and the other two sides.

It is equivalent to the **Parallelogram Law of Elementary Geometry.**

## Apollonius Theorem Formula

Let \(a\), \(b\) and \(c\) be the lengths of three sides of the triangle and \(d\) be the length of a median that bisects side \(a\). Let \(m = \frac{a}{2}\), be the length of the bisected half.

Then, Apollonius Theorem states that:

\(c^2 + b^2 = 2(m^2 + d^2)\)

This is called the **Apollonius’ Theorem Formula**.

In other words, *the sum of squares of two sides* is equal to **twice** t*he sum of squares of the **median between them** and **half of the third side**.*

## Derivation of Apollonius Theorem

We can derive and give Apollonius Theorem proof using the **Law of Cosines**. Consider the figure below, using the same notation as previously.

Let median $d$ make angle \(\theta\) on the side facing \(b\) and angle \(\theta ’\) on the side facing \(c\).

Then, for the triangle with sides \(b\), \(d\) and \(m\), we have by Law of Cosines,

\(b^2 = d^2 + m^2 – 2md \cos{\theta}\)

Now, for the triangle with sides \(c\), \(d\) and \(m\), we similarly get,

\(c^2 = d^2 + m^2 – 2md \cos{\theta^\prime}\)

But now, \(\theta + \theta^\prime = \pi = 180^\circ\)

So, \(\cos{\theta^\prime} = -\cos{\theta}\), which we use in the equation connecting \(c\), \(d\) and \(m\).

Thus we get,

\(c^2 = d^2 + m^2 + 2md \cos{\theta}\)

Now, we add this to the first equation, with \(b\), \(d\) and \(m\), so the \(\cos\) terms cancel, giving,

\(c^2 + b^2 = 2(m^2 + d^2)\)

Which proves **Apollonius Theorem Formula**.

## Applications

Apollonius’ Theorem is a theorem in elementary geometry, similar to **Pythagoras Theorem**. It is useful to calculate the lengths of a median of a triangle.

It is equivalent to the **Parallelogram Law**, as stated before. This means that it can be used to find the lengths of *one diagonal *of a parallelogram if the* other diagonal and two sides* are given.

It is also a special case of **Stewart’s Theorem**, which deals with the more general situation of a **cevian**. A cevian, like a median, is a line segment that connects one vertice and the opposite side of a triangle. Unlike a median, however, it *need not bisect* the other side.

## Solved Examples

**Question 1.** A triangle has sides 7, 6 and 10 cm. Find the length of the median to the side of length 10 cm.

**Solution.** From the terminology given above, \(a = 10 \text{ cm}, b = 7 \text{ cm}, c = 6 \text{ cm}\).

As the 10 cm side is bisected, we have, \(m= \frac{a}{2} = 5 \text{ cm}\). Let the length of the median be \(d\).

Apollonius Theorem Formula states that,

\(c^2 + b^2 = 2(m^2 + d^2)\)

Substituting required values, we have,

\(6^2 + 7^2 = 2(5^2 + d^2)\)

Which gives,

\(36+49 = 2(25+d^2) \\

85 – 50 = 2d^2 \\

2d^2 = 35 \\

d = \sqrt{\frac{35}{2}} \\

d = 4.183 \text{ cm} \\

\)

Thus, the **length of the median** is \(d = 4.183 \text{ cm}\).

**Question 2.** A parallelogram has sides 6 cm and 4 cm. If one diagonal is 8 cm long, find the length of the other diagonal.

**Solution.** This situation is very similar to the case of a triangle. The *two sides and one diagonal form three sides of a triangle*. The other diagonal is a **median** which bisects the first diagonal. The second diagonal *is also bisected* by the first.

So we have to find length of median and multiply by 2 it to get length of second diagonal.

Using same notation as usual, we have,

\(a = 8 \text{ cm}, b = 6 \text{ cm}, c = 4 \text{ cm}, m= \frac{a}{2} = 4 \text{ cm}\).

Using the formula for Apollonius Theorem, we have,

\(c^2 + b^2 = 2(m^2 + d^2) \\

16+36 = 2(16+d^2) \\

52 – 32 = 2d^2 \\

2d^2 = 20 \\

d = \sqrt{10} \\

d = 3.16 \text{ cm} \\

\)

But the diagonal in parallelogram is **twice the length of median of triangl**e.

Thus, length of other diagonal \(d^\prime = 2d = 6.32 \text{ cm}\).

## FAQs

**What is Apollonius Theorem?**Apollonius Theorem connects the lengths of sides of triangle, with lengths of median.

In words, it states that the *sum of squares of two sides* is equal to **twice** t*he sum of squares of the **median between them** and **half of the third side**.*

Mathematically, let \(AB, AC\) be side lengths, and \(AD\) be the median to side \(BC\). We can write,

\(AB^2 + AC^2 = 2(AD^2 + BD^2)\)

This is the **Apollonius Theorem. **

**Why is Apollonius’ Theorem important?**Apollonius Theorem provides us a way of *calculating median lengths in triangles*, if we know their sides. This helps us solve the triangle.

**How is Pythagoras’ Theorem related to Apollonius’ Theorem?**Pythagoras’ Theorem can be seen as* a special case* of Apollonius’ Theorem, when we have an *isosceles triangle.*

Let \(AB, AC\) be equal sides of the isosceles triangle.

The median \(AD\) to the other side \(BC\) is also perpendicular to it, so \(\triangle ADB\) is a right triangle.

By Apollonius’ Theorem,

\(AB^2 + AC^2 = 2(AD^2 + BD^2)\)

But \(AB=AC\) so,

\(2AB^2 = 2(AD^2 + BD^2)\)

Dividing throughout by 2, we have,

\(AB^2 = AD^2 + BD^2\)

This is simply Pythagoras’ Theorem in the right triangle \(ADB\).

**How is median different from angle bisector?**The median is the line segment from one vertex to opposite side, that **bisects the opposite side. **

The angle bisector is the line segment from one vertex to the opposite side, that **bisects the angle at vertex.**

In general, they are **not the same**, unless the vertex is between equal sides, i.e. in *an isosceles triangle. *