Solution of Triangles: Apollonius Theorem

The solution of triangles is a fundamental topic in trigonometry that involves finding the unknown sides and angles of a triangle when certain sides and angles are given. One of the key theorems used in the solution of triangles is the Apollonius Theorem, which is particularly useful for solving problems related to the lengths of the sides of a triangle.

Apollonius Theorem

Apollonius Theorem is named after the ancient Greek mathematician Apollonius of Perga. It provides a relationship between the length of a median of a triangle and the lengths of its sides.

Statement of Apollonius Theorem

In any triangle, the sum of the squares of any two sides is equal to twice the square of half of the third side plus twice the square of the median to the third side.

Mathematically, if $a$, $b$, and $c$ are the sides of a triangle, and $m_c$ is the length of the median from the vertex opposite to side $c$, then the Apollonius Theorem states that:

$$ a^2 + b^2 = 2m_c^2 + \frac{1}{2}c^2 $$

Proof of Apollonius Theorem

Consider a triangle $ABC$ with sides $a$, $b$, and $c$. Let $D$ be the midpoint of side $BC$, and let $m_c$ be the median from $A$ to $D$. By the definition of a median, $BD = DC = \frac{c}{2}$.

Using the Law of Cosines in triangle $ABD$:

$$ a^2 = BD^2 + AD^2 - 2(BD)(AD)\cos(\angle BAD) $$

Similarly, in triangle $ACD$:

$$ b^2 = DC^2 + AD^2 - 2(DC)(AD)\cos(\angle CAD) $$

Since $BD = DC$ and $\angle BAD + \angle CAD = 180^\circ$, we have $\cos(\angle BAD) = -\cos(\angle CAD)$. Adding the two equations, we get:

$$ a^2 + b^2 = 2AD^2 + BD^2 + DC^2 $$

Substituting $BD = DC = \frac{c}{2}$ and $AD = m_c$, we obtain:

$$ a^2 + b^2 = 2m_c^2 + 2\left(\frac{c}{2}\right)^2 $$

Simplifying, we arrive at the Apollonius Theorem:

$$ a^2 + b^2 = 2m_c^2 + \frac{1}{2}c^2 $$

Applications of Apollonius Theorem

Apollonius Theorem is used to:

• Calculate the length of a median if the lengths of the sides of a triangle are known.
• Find the lengths of the sides of a triangle if the length of a median and one side are known.
• Derive other important results in triangle geometry, such as Stewart's Theorem.

Examples

Example 1: Finding the Length of a Median

Given a triangle with sides $a = 5$ units, $b = 7$ units, and $c = 8$ units, find the length of the median to side $c$.

Using Apollonius Theorem:

$$ 5^2 + 7^2 = 2m_c^2 + \frac{1}{2}8^2 $$ $$ 25 + 49 = 2m_c^2 + 32 $$ $$ 74 = 2m_c^2 + 32 $$ $$ 2m_c^2 = 42 $$ $$ m_c^2 = 21 $$ $$ m_c = \sqrt{21} \approx 4.58 \text{ units} $$

Example 2: Finding the Length of a Side

Given a triangle with sides $a = 6$ units, $b = 8$ units, and the median to side $c$ is $5$ units, find the length of side $c$.

Using Apollonius Theorem:

$$ 6^2 + 8^2 = 2(5^2) + \frac{1}{2}c^2 $$ $$ 36 + 64 = 50 + \frac{1}{2}c^2 $$ $$ 100 = 50 + \frac{1}{2}c^2 $$ $$ \frac{1}{2}c^2 = 50 $$ $$ c^2 = 100 $$ $$ c = 10 \text{ units} $$

Table of Differences and Important Points

Property	Apollonius Theorem	Other Theorems (e.g., Pythagorean)
Applicability	Any triangle	Right-angled triangles
Relation	Relates sides and median	Relates sides only
Formula	$a^2 + b^2 = 2m_c^2 + \frac{1}{2}c^2$	$a^2 + b^2 = c^2$ (for right-angled triangles)
Known Quantities Required	Lengths of two sides and median or one side and median	Lengths of two sides
Unknown Quantity Determined	Length of the third side or median	Length of the hypotenuse or other side

In conclusion, the Apollonius Theorem is a powerful tool in the solution of triangles, providing a method to find unknown side lengths and medians. Its applications extend beyond simple calculations to more complex geometric proofs and theorems.