Suppose ABCD is a cyclic quadrilateral and P is the intersection of the lines determined by AB and CD. Show that PA*PB=PD*PC

pliwraih

pliwraih

Answered question

2022-07-22

Suppose A B C D is a cyclic quadrilateral and P is the intersection of the lines determined by A B and C D. Show that P A · P B = P D · P C

Answer & Explanation

frisiao

frisiao

Beginner2022-07-23Added 13 answers

This is a case of the power of a point theorem. You can prove it using similar triangles. By the inscribed angle theorem, B A C = B D C and so
P A C = P D B .
Again by the inscribed angle theorem,
A B D = A C D .
By A Asimilarity, this establishes that
P C A P B D .
By similarity ratios,
P A P D = P C P B ,

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