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Problem first posed by Lucas in 1877.
Very thorough website in German:
Animation by G. Tulloue:

When n points M1,M2,,Mn (traditionally, flies, mice, ladybirds...) chase one another at the same constant speed, with Mk chasing Mk+1 (and Mn chasing M1), the trajectories of these points are mutual pursuit curves.
To get the figure opposite, we had the differential system stemming from the n relations solved by Maple.

In the case of a triangle, it can be noted that the two flies that were the furthest apart are the ones who meet first!

It can be proved that the meeting point is one of the two Brocard points of the triangle.

When the initial figure is a regular polygon with the points in their apparition order, then the trajectories are logarithmic spirals with asymptotic points the centre of the polygon.
The parameter of the spiral is , and the polar tangential angle is where n is the number of sides of the polygon.
The length of the trajectory of any fly is therefore equal to where R is the radius, and the length of the side.


Case of a pentagon
Case of a hexagon
Case of an octagon

Case of a square, copied 4 times by symmetries
Case of an equilateral triangle, copied 7 times by symmetries

See also similar curves in 3D.

Did the anonymous Ivorian artist who designed this engraving think they were tracing pursuit curves?

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© Robert FERRÉOL, Alain ESCULIER  2017