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**

**What is Co-ordinate Geometry?****Rectangular Cartesian Co-ordinates****Polar Co-ordinates****Relation between Cartesian and Polar Co-Ordinates****Distance between Two given Points****Distance between Two Points in Polar Co-ordinates****Division of Line Segment****: Internal & External****Area of the Triangle Formed by Three co-ordinate Points****Condition of Collinearity of Three Points****Medians of a Triangle are Concurrent****Apollonius' Theorem****Quadrilateral form a Parallelogram****Problems on Distance Between Two Points****Area of a Triangle Given 3 Points****Worksheet on Quadrants****Worksheet on Rectangular – Polar Conversion****Worksheet on Line-Segment Joining the Points****Worksheet on Distance Between Two Points****Worksheet on Distance Between the Polar Co-ordinates****Worksheet on Finding Mid-Point****Worksheet on Division of Line-Segment****Worksheet on Centroid of a Triangle****Worksheet on Area of Co-ordinate Triangle****Worksheet on Collinear Triangle****Worksheet on Area of Polygon****Worksheet on Cartesian Triangle**

**11 and 12 Grade Math**** ****From Apollonius' Theorem to HOME PPAGE**

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