  Suppose that the lines AB and CD intersect at a point P such that PA PB PC PD andsuch that A is between points P and B, and C is between points P and D. Prove thatpoints A, B,C,D are on one circle. You are encouraged to usepropositions from Books 1, 2, 3,& 4 from Euclid's Elements. Draw apicture to help support your proof.

Question help_outlineImage TranscriptioncloseSuppose that the lines AB and CD intersect at a point P such that PA PB PC PD and such that A is between points P and B, and C is between points P and D. Prove that points A, B,C,D are on one circle. You are encouraged to use propositions from Books 1, 2, 3, & 4 from Euclid's Elements. Draw a picture to help support your proof. fullscreen
Step 1

Given that the lines AB and CD intersects at a point P such that

Step 2

Since, there is a theorem that, “A circle always passes through three non-collinear points.”

So, draw the circle through three non-collinear points A, B and C and suppose that point D is not on the circle.

Then two cases are arising,

Case 1: When the point D is outside the circle, then the line CD intersects the circle at D’ as shown in the following diagram,

Step 3

By the diagram AP.PB=CP.PD’

But given that PA.PB=PC.PD

Then supposition t...

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