**We now want to explore a problem where we
have a given circle and a line not intersecting the circle. We
wish to be able to construct a circle which is tangent to the
line and also to the circle at a given point on the circle. Lets
begin with a picture.**

**Now we wish to construct two circles which
are tangent to the given circle a at the point on the circle b
and which are also tangent to the given line cd.**

**By carefully thinking about the drawing
we arrive at the conclusion that the center of the desired circles
must lie along the line AB. Otherwise the desired circle could
not be tangent to the given circle at B.**

**Now we must find he other determining factor
in the construction. Suddenly an insight occurs. If we construct
th perpendicular to line AB at point B, then an angle iis formed
with line CD. The center of our desired circle must also lie along
this angle bisector, and so the intersection of this angle bisector
with line AB will give us our sought after center point of the
desired circle. The construction and result are shown below:**