Creating Tangents to Circles

Date: 10/24/2007 at 09:53:53
From: Princy
Subject: Drawing tangents to circles

I know how to draw a tangent from a point (outside a circle) to the 
circle.  How can I create 4 distinct tangents common to 2 circles
(d > a + b)?

Date: 10/24/2007 at 15:19:07
From: Doctor Wilkinson
Subject: Re: Drawing tangents to circles

Thanks for the nice problem, Princy!

Let's start with a picture.  You have a circle with center A and
radius a and a circle with center B and radius b, and the distance d
between A and B is greater than a + b, so that the circles don't
intersect.  Draw the line AB.  The circles may lie on opposite sides
of a common tangent line or they may lie on the same side, and there
are two such tangents in each case.  

Let's look at the second case, and let E be the point of tangency on
the circle with center A and F be the point of tangency on the circle
with center B.  Then EF intersects AB at some point X which is not
between A and B (the case where a = b and EF is parallel to AB is
easy).  Since you know how to draw a tangent to a circle from a point
outside the circle, the problem will be solved if you can figure out
how to find the point X.  

Suppose a > b, so that B is between A and E.  Then you have a big
right triangle XAE and a smaller right triangle XBF, and these
triangles are similar since they share the angle AXE.  You also know
the side AE = a and the side BF = b and the distance d = AB.  Now
maybe you can fool around with proportions to see how to construct the
segment BX.

The other case can be solved in much the same way.  Write back if you
still need help with any of this and tell me what you've tried.

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