Assignment 7: Tangent Circles

By Victor L. Brunaud-Vega



Given a line and a circle with center K. Take an arbitrary point P on the circle. Construct two circles tangent to the given circle at P and tangent to the line.

What do we know about the problem?  One thing that probably is important is that the centers of the two circles will be part of lines perpendicular to the given line.  So, it is reasonable to think that there is a relationship between P, K, and the given line, through a perpendicular line.


Drawing a line through K, perpendicular to the given line, we determine the intersections between this perpendicular line and the given circle: points A and B.

If I draw lines through points A and P, and points B and P, I will define the intersections of these lines with the given line at points C and D.

Now, drawing a line through C, perpendicular to the given line, it will intersect the line AP at the point E.  This will be the center of one of the circles we are looking for.  If I draw a line at point D, perpendicular to the given line, it will intersect the line BP at point F, and this will be the center of the other circle we are looking for.

So, let us construct the circles, centered at points E and F, and with radius EP and EF, correspondingly.  Here is an animation to see the results and some variations.

What happened with the loci of the centers of the new circles?  I made this animation to find out that the loci of points E and F has a hyperbolic trajectory.




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