Apollonius Theorem 

Apollonius theorem is named after Apollonius of Perga, a Greek mathematician who has extensive applications in triangle problems and problems involving the medians of triangles. The Theorem is a seminal result in classical geometry that gives the relationship of triangle sides and the length of the median.

1.0Apollonius Theorem Statement 

According to Apollonius's Theorem, the sum of squares of two sides is twice the square of a median, along with half the square of the third side. It describes the relationship between the sides and median of a triangle. "A median of a triangle is a line segment that connects a vertex of any triangle to the midpoint of the opposite side." 

In brief, the Apollonius theorem statement is: 

“The sum of the squares of any 2 sides of a triangle equals to twice the square on half the third side, together with twice the square on the median bisecting the third side.”

2.0What is Apollonius’ Theorem Formula? 

The formula for Apollonius's theorem is: 

PQ² + PR² = 2PS² + ½ ​QR²

Here, 

• PQ, PR, and QR are the sides of the triangle,
• PS is the median from vertex P to the midpoint S of side QR,
• PS² is the square of the median,
• QR is the base of the triangle, and
• S is the midpoint of side QR.

3.0Apollonius Theorem Proof

Although the theorem can be proved using different methods, let us prove the Apollonius theorem with the help of  Pythagoras theorem. 

Proof by Pythagoras Theorem 

To Prove: PQ² + PR² = 2PS² + ½ QR² 

Given: In triangle PQR, PS is a median. 

Construction: Construct PT perpendicular to QR. 

Proof: PS is the median bisecting side of the QR 

QS = SR = QR/2 ……………..(1)

Applying Pythagoras' theorem in triangles PTQ, PTR, and PTS we get, 

In triangle PTQ, 

PQ² = QT² + PT²   ……………..(A)

In triangle PTS, 

PS² = ST² + PT²  ……………….(B)

In triangle PTR, 

PR² = RT² + PT² …………………(C)

Adding equation (A) and (C) 

PQ² + PR² = QT²  + PT² + RT² + PT²

PQ² + PR² = QT² + 2PT² + RT²

By using equation (B) 

PQ² + PR² = QT² + 2(PS² - ST²) + RT²

PQ² + PR² = 2PS² + QT² - 2ST² + RT² 

PQ² + PR² = 2PS² + QT² - ST² + RT² - ST² 

PQ² + PR² = 2PS² + (QT - ST)(QT +ST) + (RT - ST)(RT + ST)

Given in the figure that, RS = ST+RT, QS = QT - ST …….(D) 

PQ² + PR² = 2PS² + (QT + ST) QS + (RT - ST) RS

From equation 1 

PQ² + PR² = 2PS² + (QT - ST) (QR / 2) + (RT - ST) (QR/2)

PQ² + PR² = 2PS² + (QT × QR) /2 + (ST × QR)/2 + (RT × QR)/2 - (ST × QR)/2

PQ² + PR² = 2PS² + (QT × QR) /2 + (RT × QR)/2

PQ² + PR² = 2PS² + (QR / 2) (QT + RT)

It is known that QT + RT = QR 

PQ² + PR² = 2PS² + (QR / 2) (QR)

PQ² + PR² = 2PS² + (1/2) QR²

Hence Proved.

4.0Applications of Apollonius Theorem

1. Finding the Median Length: If you are given the sides of any triangle, you can use Apollonius's Theorem to find the length of the median AM. For instance, if you have the lengths of AB=5, AC=6, and BC=7, then you can use the theorem to determine the length of the median.
2. Solving for Missing Side Lengths: If a triangle has the lengths of the sides given, but one side is missing, the square of the median may be applied using Apollonius's Theorem to solve for the missing length.
3. Characterising Isosceles Triangles: For the isosceles triangle whose two sides are the same, the median from the vertex opposite the base is also the altitude. Apollonius's Theorem may prove to be useful in proving this symmetry.