Question

If AB and CD are two chords which when produced meet at P and if AP = CP, show that AB = CD.

Answer 1

Here, chord AB and CD of the circle intersect at P.
∴ PA x PB = PC x PD

⇒ PB = `("PC" xx "PD")/"PA"`

⇒ PB = `("AP" xx "PD")/"AP"` ...( ∵ PC = AP(Given))

⇒ PB = PD                             ...(i)
Now, AB = AP - BP
⇒ AB = CP - PD                     ...(∵AP = CP(Given), BP = PD(from (i)))
AB = CD
Hence, AB = CD
Hence proved.