If two equal chords of a circle . . .

Question : If two equal chords of a circle intersect within a circle. Prove that the segment of one chord are equal to the corresponding segment of another.

Doubt by Samridhi

Solution : 

Given :  A circle with centre O. AB and CD are two chords such that AB=CD. 

To Prove : 
(i) AP=DP
(ii) BP=CP

Construction : Draw OM⊥AB and ON⊥CD

Proof : 
(i) In ∆OMP and ∆ ONP

∠OMP = ∠ONP (Each 90°)
OP = OP (Common)
OM = ON (Equal chords are equidistant from the centre of a circle)
∆OMP≅∆ONP (By RHS)
MP=NP — (1)  (By CPCT)

AM=BM (Perpendicular drawn from the centre to the chord bisect the chord)
Similarly 
DN=CN

Also 
AB=CD — (2) (Given)
AM+BM=DN+CN
AM+AM=DN+DN
2AM=2DN
AM=DN — (3)

Adding equation (1) and (2) 
AM+MP=DN+NP
AP=DP — (4)

(ii) Subtracting equation (4) from (2)

AB-AP=CD-DP

BP=CP