Apollonius Theorem

Introduction

Whether Apollonius', Pythagoras, or any other, a theorem refers to a statement that has been proved to be true by using already known mathematical operations, facts, and arguments, like axioms. In mathematics, the set of different axioms used to prove the theorem to be correct is of numerical logic and often in the form of a question. The step-by-step process for proving a theorem to be accurate is called proof.

Apollonius' Theorem

Named after a Greek mathematician Apollonius, this theorem is an elementary theorem that relates the length of a median of any triangle to the lengths of its edges. It states that the sum of squares of any two sides of a triangle is equal to twice of its square on half of the third side, along with the twice of its square on the median that bisects the third side. Apollonius' theorem, in general, is proved to be correct by using coordinate geometry, but it can also be proved by using the Pythagorean theorem and vectors. Now, let's go through the statement and proof of this theorem.


Statement: If O is the mid-point of a side MN of any triangle LMN, then LM² + LN² = 2(LO² + MO²).


Proof: First of all, choose the origin of the rectangular Cartesian coordinates at O and x-axis along the side MN and OY as the y-axis. Now, if the side MN is equivalent to 2a, then the coordinates of M and N are (a, 0) and (- a, 0) respectively. Now, referring to the opted axis if the coordinates of L are (b,c), then

LO² = (C – 0)² + (b – 0)², because the coordinates of the O are (0,0)

        = c² + b²,

MO² = (0 – 0)² + (- a – 0)² 

        = a²,

LM² = (c – 0)² + (b + a) ² 

        = c² + (a + b)²,

And, LN² = (c – 0) ² + (b – a) ² 

                  = c² + (a – b)²

Now, on adding LN² and LM², we will get:

LN² + LM² = c² + (a + b)² + c² + (b – a)²

                    = 2c² + 2 (a² + b²)

                    = 2(b² + c²) + 2a²

= 2LO² + 2MO²

= 2 (LO² + MO²).

= 2(MO² + LO²). Hence Proved

Statement and Proof by the Pythagorean Theorem 

Statement: For a triangle ABC with M be the midpoint of its side BC, 

AB2 + AC2 = 2{AM2 + (BC/2​)2}


Proof:  Let AH be the perpendicular from A on BC

So, it is clear that

BM = CM = BC/2

BH + CH = BC

Now, use the Pythagorean Theorem

AB2 = AH2 + BH2

AC2 ​ = AH2 + CH2

AM2 = AH2 + MH2.​

From the above equations, we can conclude that:

AB2 + AC2 = 2AH2 + BH2 + CH2

                 = 2AH2 + 2MH2 + BH2 − MH2 + CH2MH2

                 = 2AM2 + (BH + MH) (BH−MH) + (CH+MH) (CH−MH)

              = 2AM2 + (BH+MH) ⋅ BM + CM ⋅ (CH−MH)

              = 2AM2 + BC2/2

              = 2 (AM2 + (BC/2)2). Hence proved

Statement and Proof by Vectors

Statement: For a triangle, ABC having M as the midpoint of side BC, AB2 + AC2 = 2 (AM2 + (BC/2)2), i.e., triangle ABC satisfies Apollonius's theorem by using vectors. 


Proof: Let A be the Cartesian coordinate of triangle ABC and define AB = ∣b∣ and AC = ∣c∣, then it is clear that AM = (b+c)/2​ and BC = ∣c∣ - ∣b∣

AB2 + AC2 ​= ∣b∣2 + ∣c∣2

                 =1/2 ​(2∣b∣2 + 2∣c∣2)

                 = ½ ​(∣b∣2 + ∣c∣2 + 2b⋅c + ∣b∣2 + ∣c∣2 − 2b⋅c)

                 = ½ ​{(b+c)2 + (c−b)2}

                 = ½ (4AM2 + BC2)

                 = 2 (AM2 + (BC/2)2) Hence Proved.

FAQ (Frequently Asked Questions)

1. What is Apollonius Theorem Formula?

A. Apollonius theorem is a theorem in Mathematics which gives the relation between the median of a triangle and its sides. The theorem is also called the median of a triangle theorem.  This theorem states any two sides of a triangle when squared individually and added, the sum will be equal to the sum obtained when two times the square of half the third side and two times the square of median from the third side to the opposite vertex are added. In general, if ABC is a triangle with sides AB, BC and AC taken in order and AD is the median to the side BC, then Apollonius theorem formula is given as:

AB2 + AC2 = 2 (AD2 + BD2)

2. What is the Converse of the Apollonius Theorem?

A. Apollonius theorem also known as Median of a triangle theorem states that in triangle ABC, if AB, BC and AC are the sides and AD is the median to BC, then AB2 + AC2 = 2 (AD2 + BD2).

Converse of this median theorem would be that if the sum of the squares on any two sides of a triangle is equal to twice the sum of squares of line joining the point on 3rd side side and the opposite vertex and the distance between one of the vertices on third side and the point of intersection of newly constructed line with the third side, then the line joining the third side and the opposite vertex is the median of the triangle. To make it more simple, the converse of Apollonius theorem is if in triangle ABC, AB2 + AC2 = 2 (AD2 + BD2), then AD is the median and BD = CD.

However, there is no practical hypothesis to prove the converse of Apollonius theorem.