Ronald Aguilar

*Given two circles and a point on one of the circles. Construct a circle tangent to the circles with one point of tangency being the designed point.*

Here in the construction of the figure that we need to solve the our problem. There is circle with center A. The circle inside with center B and a point C on circle A.

The problem gives two different solutions.

*Why?*

The circle can run through the point C with a distances closest to C on circle B, with the circle B outside of the tangent circle. Center D displays this below.

Also, The circle can run through point C with a distance farthest to C on circle B, with the Circle B inside of the tangent circle. Center D' displays this below.

A) We construct a line to go through the center A and point C.

B) Using the center C, we construct a circle. Use the radius from center B.

C1) For the 1st graph - Using the new circle C (outside circle A) we created, make a point E at the intersection of the circle and line through E.

C2) For the 2nd graph - using the the new circle C (inside circle A and circle D') we created, make a point E at the intersection of the circle and the line through E.

D) Construct a segment (pink) between E and B. The gray line is constructed as a perpendicular line to F and segment EB.

E) The center D' intersects the gray line and blue line.

Circle B outside tangent circle D

Circle B inside tangent circle D'

Moving C about the circle A. Center D will make an ellipse.