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Intersecting Chords

Date: 06/09/2002 at 15:33:36
From: Elfspirit
Subject: Geometry- urgent

Hello. Here is my math problem:

There are two chords inscribed in a circle, AB and CD. They intersect
at point E. Prove that AE * EB = CE * ED.

First, I drew chords AD and CB to create two triangles. The vertical
angles AED and CEB are congruent, and so are angles AEC and DEB. 

But I am lost from there. I know I should prove that the two triangles
formed are similar triangles, but I'm still not entirely certain how
much that will aid me in this proof. 

Is it possible the proof was phrased wrong and it means that 
AE * EB is PROPORTIONAL to CE * ED?  

Please, please, please answer as soon as you possibly can; this is
very important.


Date: 06/10/2002 at 03:52:40
From: Doctor Floor
Subject: Re: Geometry- urgent

Hi,

Thanks for your question.  Consider the following figure:

  

Indeed angles <AEC and <BED are congruent vertical angles.

Also <CAB and <CDB are congruent, because they are inscribed angles 
on the same arc. In a similar fashion <ACD and <ABD are congruent.

This shows us that triangles BED and CEA are similar (note that B 
corresponds to C and D to A). From that:

  BE/CE = ED/EA

  BE*EA = CE*ED

If you have more questions, just write back.

Best regards,
- Doctor Floor, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
High School Euclidean/Plane Geometry

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