Given two circles, one with a designated point on the outside, how can we construct a third circle that is tangent to both?

The construction involves creating a line through the center of the circle to the desired point of tangency on the outside. We then create a circle with the radius of the smaller circle that has as its center point the desired point of tangency.

We can use the center of the original inside circle and the top intersection of the new circle with the line through the center of the larger circle to create a segment. The perpindicular bisector of this segment will intersect with the original line through the center of the circle at the center of the desired tangent circle.

It is actually possible to create two circles that are tangent to the orginal circles.

We can see that the center of both circles are inside the

original circle with the desired point of tangency and that the centers are collinear with the center of the circle with the designated tangency point.

If the two circles intersect, where can we find the centers of the desired tangent circles?

Again they are collinear with the center of the circle that contains the desired tangency point.

What if the second circle lies entirely outside of the circle with the designated tangency point? We see that the centers of the tangent circles are not collinear with the center of the circle with the desired point of tangency:

Why is it that the centers are collinear in every case except for the case in which the smaller circle is completely external to the circle with the desired point of tangency? Because in the final case the radial measure for the larger tangent circle is equal almost to the diameter of the smaller tangent circle and it is almost twice the diameter of the circle with the desired tangency point. This difference in measure, while requiring still that these 3 circles meet at the same designated tangency point, requires that their centers are no longer collinear.