**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.

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