Q. Prove that the points A,B,C,D are concyclic if the line segments AB and CD intersect at a point P such that AP.PB = CP.PD
Asked by imabhi264 | 14th Apr, 2017, 12:13: PM
Construction: Draw the circle through the three non – collinear points A, B, C.
This is possible, according to the theorem that 'A circle always passes through three non-collinear points'.
If D lies on this circle, then the result follows.
A, B, C and D are concyclic.

If possible, suppose D does not lie on this circle. Then, this circle will intersect CD at D’. Join D'B.
So, AP.PB = CP.PD'
But we are given that AP.PB = CP.PD.
D' coincides with D.
D lies on the circle passing through A, B and C.
Hence, the points A, B, C and D are concyclic.
Construction: Draw the circle through the three non – collinear points A, B, C.
This is possible, according to the theorem that 'A circle always passes through three non-collinear points'.
If D lies on this circle, then the result follows.
A, B, C and D are concyclic.
If possible, suppose D does not lie on this circle. Then, this circle will intersect CD at D’. Join D'B.
So, AP.PB = CP.PD'
But we are given that AP.PB = CP.PD.


Hence, the points A, B, C and D are concyclic.
Answered by Rebecca Fernandes | 27th Nov, 2017, 01:18: PM
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