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.
● Co-ordinate Geometry