Question

$AB$ and $CD$ are two chords when produced meet at $P$ and if $AP=CP$.Show that $AB=CD$

Answer 1

Given:$AB$ and $CD$ are two chords when produced meet at $P$ and $AP=CP$ ......$\left(1\right)$

$PA\times PB=PC\times PD$

$PA\times PB=PA\times PD$ since $PA=PC$

$\Rightarrow PB=PD$ ......$\left(2\right)$

$\left(1\right)-\left(2\right)\Rightarrow AP-PB=CP-PD$

$\Rightarrow AB=CD$

Hence proved.