Intersecting Circles

Date: 05/13/99 at 17:36:45
From: Peter
Subject: Intersecting circles problem

Problem: Two circles intersect at A and B. A common tangent touches 
the two circles at S and T. Show that the line AB bisects the common 
tangent ST.

I have tried various methods, e.g. using the intercept theorem and 
circle properties, and I am still stumped. I would appreciate a 
solution. Thank you.

Date: 05/14/99 at 05:15:04
From: Doctor Floor
Subject: Re: Intersecting circles problem

Hi, Peter,

Thanks for your question!

We can use here that the power of a point with regard to a circle is 
constant. 

What is the power of a point w.r.t. a circle? Let P be a point, and let 
a line through P intersect the circle at points B and C. Then the power 
of P w.r.t. the circle is PB*PC. When P is inside the circle, the 
product is negative.

For this problem we will consider P to be outside the circle. Why 
doesn't it matter what line through P intersects the circle?

Let's look at a diagram:

  

We can see that angle PDB = 180 deg - angle EDB = angle ECB = angle
ECP, and that angle PBD = angle PEC, so triangles PDB and PCE are similar. 
Thus PD/PB = PC/PE, and PB*PC = PD*PE. We can conclude that the power is 
constant.

When the line through P is tangent to a circle, say at point G, then of 
course the power of P w.r.t. the circle becomes PG*PG = PG^2.

Now, for your problem let us consider a tangent common to two circles, 
and the line connecting the intersection points of these circles:

  

The power of S w.r.t. the circle with center A is: SR^2 = SF*SE.
The power of S w.r.t. the circle with center B is: SP^2 = SF*SE.

So we have SR^2 = SP^2, and hence SR = SP, as required.

If you have a math question again, please send it to Dr. Math!

Best regards,

- Doctor Floor, The Math Forum
  http://mathforum.org/dr.math/   

Date: 05/14/99 at 22:52:12
From: Peter Ooi
Subject: Re: Intersecting circles problem

Thank you, Dr. Math, for your solution to the Intersecting Circles
problem. My students will appreciate the solution. I knew the theorem ?
or theory of power of a point w.r.t a circle was not specifically
mentioned in the New South Wales HSC Maths syllabus but I can certainly
make changes to the question to suit the syllabus. 

Thanks again and best regards from a teacher from Oz.