Question

prove that the points A,B,C and D are concyclic if the line segment AB and CD intersect at point P such that AP.PB=CP.PD

Answer 1

Answer:

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.

rightwards double arrowD' coincides with D.

rightwards double arrowD lies on the circle passing through A, B and C.

Hence, the points A, B, C and D are concyclic.

Step-by-step explanation:

Answer 2

Step-by-step explanation:

answer is in the attachment