Geometry Problem

Click the figure below to see the complete problem 536 about Intersecting Circles, Chord, Perpendicular.

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## Wednesday, November 3, 2010

### Problem 536: Intersecting Circles, Chord, Perpendicular. Level: High School, SAT Prep, College Geometry

Labels:
chord,
intersecting circles,
perpendicular

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extend AC to G on circle O, => BCG isoceles

ReplyDelete=> AD = DC + CG = BC + BC

Let E is the point of intersection of AC to circle O.

ReplyDeleteConnect OO’ and note that OO’ is the perpendicular bisector of AB

1. We have Angle(BEC)=angle (O’OA)

And angle(ACB)=2 *angle(O’OA)=angle(BEC)+angle(CBE)

So angle(CBE)=angle(O’OA)=angle(CBE)

Triangle BCE isosceles and CB=CE

2. Since AE is a chord of circle O so D is the midpoint of AE

We have Area(ADO)=Area(DOC)+Area(COE)

All above triangles have the same height so AD=DC+CE

Or AD=BC+CB

Peter Tran