**Apollonius’ theorem** is an elementary geometry theorem relating the length of a median of a triangle to the lengths of its sides. It states that **“the sum of the squares of any two sides of any triangle equals twice the square on half the third side, together with twice the square on ****the*** median bisecting the third side”. *Apollonius’ theorem is proved by using co-ordinate geometry. Proof of this geometrical property is discussed with the help of step-by-step explanation along with a clear diagram.

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

**Proof:** Let us choose origin of rectangular Cartesian co-ordinates at O and x-axis along the side MN and OY as the y – axis. If MN = 2a then the co-ordinates of M and N are (- a, 0) and (a, 0) respectively. Referred to the chosen axes if the co-ordinates of L be (b, c) then

LO² = (b – 0)² + (C – 0)², [Since, co- ordinates of O are (0, 0)]

= b² + c²;

MO² = (- a – 0)² + (0 – 0)² = a²

LM² = (b + a) ² + (c – 0)² = (a + b)² + c²

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

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

= 2(a² + b²) + 2c²

= 2a² + 2(b² + c²)

= 2MO² + 2LO²

= 2(MO² + LO²).

= 2(LO² + MO²). *Proved.*

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