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
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