Circle Geometry

Date: 6/4/96 at 17:29:36
From: Anonymous
Subject: Circle Geometry

Two circles intersect at A and N. One of their common tangents has 
points of contact P and T. Prove that <PAT and <PNT are supplementary.

I know that 'supplementary' means that the angles have to add up to 
180 degrees.  

I don't know what to do becuase there are 2 circles, so you can't use 
other theorems like 'Tangent Cord Theorem' or 'Tangent Theorem'.

Thank you very much.

Sheryl Korenzvit

Date: 6/5/96 at 17:3:3
From: Doctor Ceeks
Subject: Re: Circle Geometry

You want <PAT + <PNT = 180, which is equivalent to showing
<PAT+<P'AT '= 180 where P' and T' are the points of contact of the
other common tangent.  But this is equivalent to showing
<PAP' + < TAT'=180.  Now notice that the radii through P and T are
parallel since both are perpendicular to PT.  Also the radii through 
P' and T' are parallel since both are perpendicular to P'T'.  We 
conclude that the arc PP' not through A and the arc TT' not through A 
together measure 360, and the problem follows.

Please write back if you need more explanation.

Doctor Ceeks,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/