Math CentralQuandaries & Queries


Question from Jacob, a student:

If three circles are circles, does that mean that the two tangent lines are perpendicular?

NO. Circles are "mutually tangent" when each pair of them touch at a single point. At that point their common tangent will be perpendicular to the line that joins their centers. There are two ways that three circles can be mutually tangent:

  1. Start with triangle ABC and the points where the incircle touches the three sides -- D on BC, E on CA, and F on AB. (That means you must bisect two of the angles, and call I the point where the bisectors meet. Drop a perpendicular from I to D on BC, to E on CA, and to F on AB.) Then the circles with center A and radius AE = AF, center B and radius BF = BD, and center C and radius CD = CE will be three mutually tangent circles.

  2. Their centers can lie on a line. In this case the centers can be any points along the line, and you draw the circles so that they pass through a common point P of that line.


About Math Central


Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.
Quandaries & Queries page Home page University of Regina PIMS