Apollonius' Theorem



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).

Apollonius theorem

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
LMB2 = (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

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