Given two circles and a point on one of the circles. Construct a circle tangent to the two circles with one point of tangency being the designated point.
LetŐs start with one large circle with center A and construct a smaller circle inside the larger circle, center B.
Then, place an arbitrary point on each circle, C and D respectively.
Next, construct a line segment from point B to point C, this will be the radius of a circle with center D; additionally, create a line that goes through points A and D.
Then, use the line segment BC to create a like circle with center D.
Next, label a point, E, lying on the newly constructed circle and also on the new line.
Then, construct a line segment from point B to point E and label the midpoint F.
Next, construct a line perpendicular to segment BE that passes through point F. This line passes through line AD at newly labeled point, G.
Finally, create a circle with a radius equal to the length of GD which creates our circle that is tangent to the two other circles.