Question

$AB$ and $CD$ are two chords of a circle intersecting at $P$. Prove that $AP\times PB=CP\times PD$.

Answer 1

To Prove : $AP\times PB=CP\times PD$

Construction : Join $AD$ and $CB$

Proof : In $△APD$ and $△CPB$

$\angle A=\angle C$ [angles of the same segment]

$\angle C=\angle B$ [angles of the same segment]

$\therefore$ $△APD\sim △BPC$ [AA Postulate]

$\therefore$ $\frac{AP}{CP}=\frac{PD}{PB}$

[corresponding sides of similar $△$s are proportional]

$\therefore$ $AP\times PB=CP\times PD$