Tangent Circles

By: Lacy Gainey


Given 2 circles, A and B, we can find a third circle, C, that is tangent to both of these circles.
Lets start off by constructing two circles.

Next, place a point on circle A.  Label this point K.
Construct a line through the center of circle A and point K. Label this line L.

At point K construct a circle with the same radius as circle B.  Construct a point at the intersection of circle K and line L.  Label the point M.

Construct a line segment from point M to the center of circle B.  Find the midpoint of this segment and label it N.

Construct a line though point N that is perpendicular to the line segment BM.

Construct a point at the intersection of the perpendicular line and line L.  Label this point O.

Lastly, highlight point O and point k and construct a circle using the center + point command. 

Circle O is tangent to both circle A and circle B.

Click here for a GSP script tool for the construction of this tangent circle.

Lets look at constructing a tangent circle when circle A and circle B have different characteristics.

Case 1: When circle B is lies completely inside of circle A.
This case is represented in the construction we did above.

When circle B is completely inside of circle A, the tangent circle O is inside of circle A as well.

Case 2:  When circle A and circle B overlap.

When circle A and circle B overlap, the tangent circle O is still contained within circle A.

Case 3:  When circle A and circle B are disjoint.

When the two circles are disjoint, circle A is contained within the tangent circle O.

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