Tangent Circles

Write-up by

Blair T. Dietrich

EMAT 6680

Assignment #7

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.

The construction:

Given circles O and B and a point A on circle O.  Construct the line containing O and A.

Construct a circle centered at A that has the same radius as circle B.  Circle A intersects line OA in two places.  Label the outside point of intersection C and constuct the segment from B to C.

Construct the perpendicular bisector of BC.  This line intersects OA at point D.

The distance from D to A is the same as the distance from D to the edge of circle B.  By constructing a circle centered at point D that passes through A, we have a circle that is tangent to both circle O and circle B.

Investigations:

MORE CIRCLES

When point A travels around the edge of circle O, the midpoint of segment BC used in the construction traces out a circle (shown in blue).  This is true regardless of the position of circles O and B.

ELLIPSES and HYPERBOLAS

When point A travels around the edge of circle O, the center point of circle D traces out an ellipse (shown in blue).  This will always be true as long as circle B lies entirely or partially inside circle O.

However, if circle B is completely outside of circle O, the locus of points traced by point D is a hyperbola.

By definition, the sum of the distances from a point on an ellipse (point D) to each of the two foci (points O and B) has to remain constant.  Also, the magnitude of the difference between the distances from a point on a hyperbola (D) to each of the focal points (O and B) must remain constant.  Notice these relationships hold true as you investigate here.

(A sample screen shot is shown below.)