Get the answer to "Suppose A B C D is a cyclic..."

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 \cdot P B = P D \cdot P C

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 .

A A

similarity, this establishes that

△ P C A \sim △ P B D .

By similarity ratios,

\frac{P A}{P D} = \frac{P C}{P B},

Do you have a similar question?

Recalculate according to your conditions!

Didn't find what you were looking for?