AB and CD are two chords of a circle intersecting each other at P such that AP = CP. show that AB = CD
(1) $\displaystyle AP=CP$..........................................Given
(2) $\displaystyle AP \cdot PB = CP \cdot PD$.............................When chords intersect in a circle, the product of their segments is equal.
(3) $\displaystyle PB = PD$..........................................Division (AP=CP)
(4) $\displaystyle AP+PB = CP+PD$...........................Addition (1) and (3)
(5) $\displaystyle AP+PB=AB \ \ and \ \ CP+PD=CD$.......Segment Addition Postulate
(6) $\displaystyle AB=CD$..........................................Substitu tion (4) and (5)