Oct. 11th, 2010

I've been spending the past few days playing with CaRMetal, which is an awesome application for generating mathematical figures. Here's a problem that I just finished working on.

In this diagram, the three internal circles all have the same radius and share the same point Q. ABC is a triangle "circumscribing" the three circles in the sense that each side is the unique line tangent to two of the three circles without crossing the third circle. R is the center of the circle that passes through the three vertices of the triangle (partially shown), and P is the center of the circle that is tangent to the three sides of the triangle (which isn't shown). In the diagram shown, P, Q, and R are on the same line, or at least they appear to be. The homework problem that was associated with this diagram was showing that this was the case, and that it works out this way no matter how you choose three circles of the same radius that share a point.

(For the record, I barely have the first clue of how you would show this to be true. I had some nasty punitive homework in college.)

The thing that makes CaRMetal a home run is that this diagram wasn't drawn so much as constructed. That line AC isn't "eyeballed" to sort of look like the tangent to those two circles; there are a whole structure of hidden points and lines that are all designed as being the perpendicular of this line through that point or the intersection that that line with the other circle and so on, so I can have confidence that the diagram is drawn "to scale". The thing that Makes CaRMetal a grand slam is that inside the application I can click and drag those three circles around and all of the dependent lines and points will recalculate themselves, so I can swing the circles around and watch the line segment PR move around in real time but still always pass through Q. Whoa.

So I've been rather giddily making diagrams to supplement my old college math homework. It might not turn out to be the best tool to draw graphs, but at least I should be able to draw proper pentagons instead of hand-drawn misshapen messes. Shiny.


Matthew Daly

December 2012

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