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 LM^{2} + LN^{2} = 2(LO^{2} + MO^{2}).

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 LO^{2} = (b - 0)^{2} + (C - 0)^{2} , [Since, co- ordinates of O are (0, 0)] = b^{2} + c^{2}; MO^{2} = (- a - 0)^{2} + (0 – 0)^{2} = a^{2} LMB^{2} = (b + a) ^{2} + (c – 0)^{2} = (a + b)^{2} + c^{2} And LN^{2} = (b - a) ^{2} + (c - 0) ^{2} = (a - b)^{2} + c^{2} Therefore, LM^{2} + LN^{2} = (a + b) ^{2} + c^{2} + (b - a)^{2} + c^{2} = 2(a^{2} + b^{2}) + 2c^{2} = 2a^{2} + 2(b^{2} + c^{2}) = 2MO^{2} + 2LO^{2} = 2(MO^{2} + LO^{2}). = 2(LO^{2} + MO^{2}). Proved.