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

beefypy 2022-10-25 Answered
Suppose ABCD is a cyclic quadrilateral and P is the intersection of the lines determined by AB and CD. Show that P A · P B = P D · P C.
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Answers (1)

dkmc4175fl
Answered 2022-10-26 Author has 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 P B D ..
By similarity ratios, P A P D = P C P B , which is what we wanted.
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