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).

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 

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