Solved: "Suppose ABCD is a cyclic quadrilateral and P..."

beefypy

Answered question

2022-10-25

Suppose ABCD is a cyclic quadrilateral and P is the intersection of the lines determined by AB and CD. Show that

P A \cdot P B = P D \cdot P C

dkmc4175fl

Beginner2022-10-26Added 15 answers

Step 1
This is a case of the power of a point theorem. 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 .

.
Step 2
By AA similarity, this establishes that

△ P C A \sim △ P B D .

.
By similarity ratios,

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

which is what we wanted.

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