Problem Solving: Part 1 of 2. Do the Problem!

Hey, so this post is another digression from the deep and depressing topic of education reform.  I saw this question posted on Twitter a few days ago:  How do you construct a circle tangent to three other circles that do not share any disc space (circle interiors don’t overlap at all)?  (Shout out to John Golden @mathhombre for posting the problem)

Took me a couple of days of messing around with GeoGebra (and putting up with my wife’s smirks) to figure it out.  What I love about problems like this is that it replicates what I think mathematics ought to be like: we have a problem which may or may not be based on a real life scenario.  I would be hard pressed to come up with a real-life scenario for this particular problem.  I saw this as a brain teaser, more or less just for fun.  It took me a few tries, a pretty good investigative tool (GeoGebra), and some mental soak time to get there, but I really enjoyed conquering this problem.

My initial approach was a classic one for solving math problems:  start with a simpler problem.  I started with two circles instead of three. I drew some different-sized circles, and manually moved them so that they were tangent to the other two circles, and tried to see if there was a pattern in the locations of the centers of the circles.  By squinting at the drawing (one of my favorite geometry techniques) it looked like the centers followed a curved pattern:

Aha, a curve!  Given the setting, my index of suspicion was high for a conic.  The “Conic through 5 points” tool in GeoGebra came in handy here.  Pick any five of the circle centers and you get a hyperbola!  All the centers lie on one branch of the hyperbola, and that branch also goes through the last center.  Further, if you start a new circle anywhere on that branch it’s tangent to the two circles!

Aha!  Now we’re getting somewhere.  The hyperbola tool in GeoGebra requires as input the two foci and any one point on the curve.  Simple examination of the drawing shows that one of the circles tangent to both circles is centered on segment between the two circle centers, midway between the two circle paths.  If you construct the segment from one center to the other, construct points where that segment intersects the two circles, and then construct the midpoint of those two points, that point must lie on the hyperbola:

What about the foci?  I took a guess and used the centers of the circles as the foci.  Once I constructed the point halfway between the two circles (point D in the drawing above), you can use the hyperbola tool to construct the conjectured hyperbola in blue below:

Pretty close!  The left branch is off a little, but I chalked that up to the “sensitivity” of the locations of my original trial circles that I just “eyeballed” into tangency.  I like the blue hyperbola, and I think it will work.

Complication:  depending on the relative size of the circles (which one’s bigger than the other), the locus of tangent circle centers switches from one branch of the hyperbola to the other.  I can foresee some complications arising when dealing with three circles.

Now to try with three circles.  In a new window, I drew three different sized circles, created three sets of hyperbolas using each two out of the three circles at a time:

Nice!  Notice that all three hyperbolas actually intersect at two points on the drawing.  One appears to be the center of the circle that would be externally tangent to all three circles.  Construct the intersection and draw a circle….

Yup!  What about the other intersection point?  Wait a minute, it looks like, maybe….

Hey!  It’s the center of the circle that is internally tangent to all three circles!  Now THAT’S Math porn, baby!

As you fiddle around with the circles and make them different sizes relative to each other, or try to make them the same size, some weird things do happen, and the circles drawn here don’t end up being the tangent circles.  However, the centers of the tangent circles are always at the intersection of all three hyperbolas.  It’s just that the intersections points jump around a bit.

Anyhow, hope you enjoyed the digression into Geometry.  Check out part 2 of this post when I reflect a little more deeply on the problem solving process itself, and how it feels to me to be a problem solver.