**Tangent Circles Investigation**

**By**

**Cassian Mosha
**

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.

Given
two circles that are tangent to each other then they have a common point. Now
letŐs consider two circles as shown below.

**
Figure 1**

We will try to construct circle C that is tangent to both
circles A and B. To do as such we need to pick two points one on each circle.
The point on circle A will be an arbitrary point that we will use to construct
a line through center A. Also the arbitrary point on circle B will connect to
the center of circle B and will form a radius that we will use construct our
tangent circle to both circles**.**

**
Figure 2**

Then we will connect points E and D to form a triangle BDE.
Then we will find the mid point of segment BE and construct a perpendicular
line through this point. The intersection point on the perpendicular line L2
through line segment BE and line L1 is the locus point of the two given circles
A and B. This locus point will be the center of the circle that is tangent to
both circles**.**

**
Figure
3**

As we can see circle C is tangent to both circles A and B and if we animate point E we can describe some different features that the circles are showing while revolving at its tangential points.

** Figure
4**

Figure 4 shows the tangential circle to both circles A and B and they are equidistant from the locus. No matter where the circles are the distance between the centers of circle A and B will always beequidistant from the locus. The locus is in line with the centers of the circles A and B since any line tangent to a circle is by definition perpendicular to the radius or diameter of the circle. The locus point when animated makes a small circle that is tangent to circle B and then makes another big circle that is partly overlapping with circle B and tangent to center A.

After constructing the tangent circle lets explore how to construct two circles tangent to a given circle at point P on the circle if we are given a line and a circle with center K.

**
Figure 5**

Using the tangent circle we constructed earlier, and using the observation of the animation and the fact that a tangent line is perpendicular to a radius, I constructed the circle first, then I chose a point on the circle construct a perpendicular to form the line. Then with the given line I took a segment of radius R and draw two circles using each end point as a center. The two circles gave me two new points, which I used to construct the circles that are tangent to each other, and tangent to the given line. The circles and the line intersect at point P as shown on figure 5.