Tangent Circles and Their Loci

By

Audrey V. Simmons

For any two given circles and a point on one of the circles, a circle tangent to the two circles at the given point can be drawn.

Below is an example of two tangent circles drawn to the given green circles.  The red circle is inside the large green circle (tangent at the given point) but outside the small green circle.  The little green circle and the red circle are externally tangent.

The purple circle is tangent to the large green circle (at the given point) but its tangency is outside the little green circle.  The little green circle and the purple circle are internally tangent.

Given Circles

To use the tangent circle tool click HERE

# Loci of the centers of the tangent circles

If we look at the loci of the centers of the red tangent circle, what shape would be graphed?  Click HERE to see.

If we look at the loci of the centers of the purple tangent circle, what shape would be graphed?  Click HERE to see.

Each set of loci makes a separate ellipse.

Loci of the base of the isosceles triangle

Let’s look back at the construction of the red tangent circle, the one that

was externally tangent to the little green circle.  Segment AB is the base of an isosceles triangle.  All points on the dotted red line are equidistant from both point A and B since the line is the perpendicular bisector of AB.  Look at the locus of the midpoint of the segment by clicking HERE.

Instead of making an ellipse the loci of the midpoints is a ________? Circle!  The locus of the midpoints of the purple circle is also a circle.

Isn’t it amazing that there is such consistency in the arrangement of points.  In summary, the loci of the centers of the tangent circles form ellipses. The loci of the midpoints of the bases of the isosceles triangles (used during construction) form circles.

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