A Circle – or not?

In this video by H. Bortolossi I found an interesting problem. I do not understand all of the Portuguese, but the video was interesting nevertheless for me. Also it was nice to see my friend from Rio in action. This reminds me of my time in the most beautiful town of the world. Ah no – maybe that title goes to Venice, but it is close.

Look at the following construction.


The two blue segments are of equal length. The black and the green curves are circles with centers C1 and C2. The red curve is the track of M, when A walks round the black circle. Is the track a circle too?

The construction is done with the automatic track feature of C.a.R. The track has been made a construction object. We can then move the circles with the right mouse button. If we do that only a little bit, the track will look like a circle. But if we extend the movement, we get the following.


This speaks against the theory of a circle. The algebraic type of equations should remain the same, even in special cases. We can also check by putting three points on the track and constructing the circle through these points.


We need to carefully select a situation, where the cyan circle disagrees visually with the track. Here is magnification.


The difference is too large to be accounted to numerical errors. In fact, the track is not a circle. I did not investigate why it is so close to a circle. But I computed the equation of the track with Euler Math Toolbox. The equation of the line I got was


for a special case of the two circles. You can eliminate the square roots, and get an equation of degree 6, which does not factor. If the track was a circle, this circle should factor out. If you wish to try and see, paste the following commands into Euler Math Toolbox.

>eq &= (x-2)^2+y^2-10
>eq1 &= eq with [x=x*(3-2/sqrt(x^2+y^2)),y=y*(3-2/sqrt(x^2+y^2))];
>eq3 &= num(factor(expand(eq1)));
>$eq3=0 // needs Latex
>h &= 12*y^2+12*x^2-8*x;
>eq4 &= expand((eq3+h)^2-h^2)
>plot2d(eq3,level=0,a=-2,b=6,c=-4,d=4,levelcolor=green,n=100); ...
>plot2d(eq,level=0,>add,levelcolor=red); ...

There seems to be only the special case of concentric circles, where the track is indeed a circle.

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