[Maths Class Notes] on Apollonius Theorem Pdf for Exam

It is understood that theorems in mathematics refer to the statement of a result that has been proven based on previously set statements, such as theorems, or statements commonly acknowledged, such as axioms. In mathematics, theorems are defined as the results derived from a set of axioms that are found to be accurate. There are some axioms in mathematics that are considered mathematical logic, but with systems formulated as questions or statements.

Apollonius’ Theorem

Named after the 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 and Proof of Apollonius’ Theorem

There is no doubt that medians make up the most important set of components in the geometry of triangles and they can be regarded as independent of the geometric shapes of the triangles. According to Apollonius’ Theorem, the sides and the medians of the triangle are related. Apollonius’ theorem refers to the relationship between the lengths of the sides of a triangle and the length of its median.

Apollonius’ Theorem Statement

“The sum of the squares on two sides of a triangle equals the sum of the squares on one half of the third side, plus the sum of the squares along the median of the third side”

OR

The midpoint of any triangle LMN is O, so the formula (LM)²+ (LN)²= 2 [left [ (LO)² + (MO)² right ]] = O.

Apollonius’ Theorem Proof

(the )

Consider that O is the origin of the rectangular form and MN and OY represent the x-axis and y-axis, respectively, of the Cartesian coordinates. When MN = 2a, the coordinates of points M, as well as N, are (a, 0) and (-a, 0), respectively. When L coordinates are (b, c), then

Since the coordinates of the point O are [0, 0], LO2 = (c – 0)2 + (b – 0)2 .

= c² + b²;

In other words, LM2 = (c – 0)² + (b + a)² = c² + (a + b)²

MO² = (0 – 0)² + (- a – 0)² = a²

LN² = (c – 0) ² + (b – a) ² = c² + (a – b)²

In other words, 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 + CH² − MH²

      = 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.