Date: Thu, 2 Apr 1998 11:56:20 -0600 (CST)
From: Dave Rusin <rusin>
To: ck6@evansville.edu
Subject: Re:  spelling

>spelling of Apollonius
Oopsie! Thanks.

>construction of a circle tangent to two given lines and a circle?

A circle of radius  R  which is tangent to two lines has its center a
distance  R  from each line; in particular, the center must lie on the
bisectors of the angles between those lines. 

If this circle is also tangent to a circle of radius r then the
distance between the centers has to be | R +- r |, that is, it's
more/less than the distance to the given center by exactly r. You can
say it differently: take that angle bisector and translate
perpendicularly by a distance r to get a new line. Then this new
line's distance from the desired center will have to equal to given
center's distance from the desired center. Aha! Now we recognize
the description of a parabola.

So the center of the desired circle is on the intersection of a parabola
and a line (actually one of several parabolae/lines). Given the center,
of course, the radius  R  is determined by the tangency to the given circle.

dave