Different Cases of Tangent Circles

By Paulo Tan

 

I investigated three different constructions of a tangent circle to two given circles.  The first case starts with one circle inside the other.  The second case starts with two intersecting circles.  The last case starts with two disjoint circles.  All three cases involve d very similar constructions.  I also looked at the possible number of tangent circles in each case. 

 

The construction of the first case is a familiar one.  Start with circle C inside of circle A, pick a point of tangency on circle A (call this point G), draw a line through point A and G, and draw a circle centered at point G with the same radius as circle C. The next decision affects how the tangent circle will behave.   We look for the intersection of line AG and circle G.  There are two such intersections, point F and point H (fig1).  We initially picked point F, constructed a segment from point C to point F, found the midpoint of line segment CF, and constructed the perpendicular line to line segment CF through the midpoint of line segment CF. Point E represents the intersection of line AG and the perpendicular line to line segment CF. Thus, point E represents the center of the tangent circle. 

 

Fig1.

 

Suppose we chose point H instead of point F as the intersection of line AG and circle G.  The tangent circle now encircles circle C (fig2).  This tangent circle will encircle circle C no matter where initial choice of point G lies on circle A.  The reason is that, in this case, the locus of E is an ellipse that is always inside of circle A (fig2b). 

 

 

 

 

Fig2.

 

Fig2b.

 

Hence, in the case where we are given one circle inside another, we could always construct two distinct tangent circles. 

 

The second construction starts with circle C and circle A intersecting.  The construction is similar to the first case.  Once again, the decision to connect point C to point H or F gives two different types of tangent circles (fig3).

 

Fig3.

 

Joining point C to point H produces tangent circle EÕ.  Once again this tangent circle appears to encircle circle C.  However, if we were to choose our initial point G closer to circle C, weÕll notice that this tangent circle no longer encircles circle C.  The reason is that in this case where we start with two intersecting circles, it is now possible for line m and line p to be perpendicular.  As a result lines m and n are parallel (fig4). Remember line n represents the line through the center of circle A and point G (the point of tangency).  Line m is the perpendicular bisector of line CH.  Thus, when the initial point of tangency is chosen such that line n is parallel to line m, then tangent circle EÕ cannot exist. 

 

On the other hand line p and line n could never parallel in this case.  Therefore, tangent circle E always exists in the case of two intersecting circles. 

 

Fig4

 

 

Also when the intersection of line m and line n is outside of circle A, then tangent circle EÕ will not encircle circle C (fig5).

 

Fig5.

 

The intersection points of circle A and circle C also provide some additional insight.  At these intersection points tangent circle E and EÕ are the same.  More specifically they are the same point.  

 

Thus, in the case where two intersecting circles are given, we can hypothesize:

1.    It is not always possible to construct two distinct tangent circles.

2.    Circle EÕ may not always exist.

3.    There are two points where E and EÕ are the same. 

 

Next letÕs look at the locus of E and EÕ(fig5b). 

 

Fig5b.

 

The locus of E and EÕ confirms our hypothesis in the case of two intersecting circles.  The locus of E is again an ellipse, which means that tangent circle E will always exist.  The locus of EÕ, on the other hand, is a hyperbola.  This confirms that it is possible for circle tangent EÕ to not exist.  In addition, the two intersections of circle A and circle C coincides with the intersections of locus E and EÕ.  These intersections represent the two points where tangent circle E and EÕ are the same. 

 

 

Lastly, when we construct the tangent circle of two disjoint circles.  The construction is similar to the first two cases and we could once again have two possible tangent circles (fig6). 

 

 

Fig6.

 

As with the previous case the intersection of line m and n could be outside of circle A, which implies tangent circle EÕ will not always encircle circle C.  Also line m and n could still be parallel.  Unlike the previous case, however, it is possible for line n and line p to also be parallel (fig7).

 

Fig7.

 

So, when line n and line p are parallel tangent circle E does not exist.

 

Thus, in the case where two disjoint circles are given we hypothesize:

1.    It is not always possible to construct two distinct tangent circles.

2.    Tangent circle E and tangent circle EÕ may not exist in certain situations.

3.    It is not possible for E and EÕ to be the same.

 

Once again, weÕll investigate the locus in the case of two disjoint circles (fig7b).

 

Fig7b.

 

This time the locus E and EÕ both form a hyperbola.  Hence, our hypothesis is correct that both tangent circle E and EÕ do not exist at certain locations.  Also, since the hyperbolas donÕt intersect it is not possible for tangent circle E and EÕ to be the same.

 

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