Learning Tangent Circles

By: Russell Lawless

For me, learning how to do tangent circles has been a very hard process because I have always had trouble constructing them. I feel that many other students out there have the same problem as me. So I decided to make a 10 step process of constructing a circle that is tangent to two circles in a step by step format. At the bottom I have attached a GSP file that has the finished product.

Step 1: Construct a circle A. Put a separate point B on any part of the circle. Point B will be the point of tangency on circle C. (This is so that the animation process goes better)

Step 2: Construct circle A' with it being inside circle A.

Step 3: Make a radius for circle A'.

Step 4: Construct a circle B that has the same radius as circle A'. (You can do this by clicking on the radius and point B and selecting Construct -> Circle by Center + Radius)

Step 5: Construct a line that goes through point B and point A. (Make sure that it is a line or other cases of circle tangency will not work).

Step 6: Construct line segment A' C.

Step 7: Create the midpoint D of line segment A'C.

Step 8: Create a perpendicular bisector of line segment A'C through its midpoint D.

Step 9: Create point E where the perpendicular bisector of line segment A'C meets line AB.

Step 10: Create circle E such that the radius is the length of EB. This circle that you created is the tangent circle of two circles.

Finished Product

While exploring the explorations I found out how to create an ellipse, circle, and a hyperbola through animation and tracing.

When the center A' lies on the center A , we get a circle traced from center E. GSP

When circle A' lies within circle A, we get an ellipse when center E is traced. GSP

When circle A' lies partly inside and outside circle A, we see that an ellipse is traced by center E. However, wherever the circle A' leaves circle A so does center E. GSP

When circle A' lies outside of circle A, a hyperbola is formed by the trace of center E. GSP