Question

two equal chords ab and CD of a circle when produced intersect at a point P prove that PB is equal to BD​

Answer 1

given:AB=CD

to prove:PB=PD

const: draw OE and OQ perpendicular on AB and CD respectively

proof:

given AB and CD are two equal chords of the same circle.

OE=OQ(equal chords of a circle are equidistant from the center.)

now in triangle OEP and OQP,

OE=OQ

OP=OP(common)

angle OEP = OQP =90 degree,by construction

therefore triangle OEP = OQP (RHS congruency)

EP = QP (CPCT)

also AE=EB=1/2 AB and CQ=QD=1/2 CD (the line joining the center of the circle is perpendicular to the chord and bisects the chord.)

Now AB = AC implies AE = EB= CQ=QD ....(1)

therefore EP-BE =QP - BE

EP - BE = QP - QD (FROM 1)

BP = PD

hence proved