If two equal chords of a circle intersect within the circle, prove that the segments of one chord are equal to corresponding segments of the other chord.

Construction: Draw OL i:tnd OM perpendiculars to chords AB and CD respectively. Join OP. To prove:AP = DP and PB = CP. CIRCLES 153 Proof: Consider triangles OLP and OMP, OL = OM [Equal chords AB and CD are equidistant from the centre of the circle] OP is common. LOLP = LOMP :. AOLP:::: AOMP .. LP = PM Also, AL = LB [90° each] [RHS] ..•(i) [CPCT] ...(ii) [Perpendicular from centre to the chord bisects the chord] CM = DM ...(iii ) [Reason same as above] AL + LP = DM + MP [From (i), (ii), (iii)] AP = DP ...(iv ) Now, AB = CD AP + PB = CP + PD AP + PB :: CP + AP PB = CP. [From (iv)]