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: