Yahoo Answers: Answers and Comments for 2 circles intersect at 2 pts A&B. PQ is chord of 1st circle,RS chord of 2nd s.t. they intersect at pt E on seg AB. Show quad PRQS is cyclic? [Mathematics]
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From mina
enUS
Fri, 17 Jun 2016 05:02:28 +0000
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Yahoo Answers: Answers and Comments for 2 circles intersect at 2 pts A&B. PQ is chord of 1st circle,RS chord of 2nd s.t. they intersect at pt E on seg AB. Show quad PRQS is cyclic? [Mathematics]
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https://answers.yahoo.com/question/index?qid=20160617050228AAZP4Lu
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From Learner: i) AB & PQ are both chords of first circle...
https://answers.yahoo.com/question/index?qid=20160617050228AAZP4Lu
https://answers.yahoo.com/question/index?qid=20160617050228AAZP4Lu
Fri, 17 Jun 2016 11:39:28 +0000
i) AB & PQ are both chords of first circle intersect at E.
So by two chords intersecting theorem of a circle, PE*EQ = AE*EB  (1)
ii) Similarly, from circle (2), we have, RE*ES = AE*EB  (2)
iii) Thus from (1) & (2): PE*EQ = RE*ES
==> PE/ES = RE/EQ
Also, <PES = <REQ [Vertically opposite angles]
So ΔPES is similar to ΔREQ [SAS similarity axiom]
==> <SPE = <QRE
As these two are angles of the same segment SQ and being equal,
Points P,R,Q,S lie on the same circle,
So PRQS is a cyclic quadrilateral.