## Dividing a Segment into N parts:

Besteman's Construction

A Mathematical Droodle: What Is This About?

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Copyright © 1996-2018 Alexander Bogomolny## Dividing a Segment into N parts:

Besteman's Construction

In the summer 1995, two high school students, David Goldenheim and Dan Litchfield surprised their teacher, Charles Dietrich, with a straightedge and compass construction of 1/N-th of a given segment different from the well known construction by Euclid. The boys' use of the Geometer's Sketchpad made their discovery all the more exciting for the teaching profession. The event certainly made waves. A Chinese teacher, who along with his students found a generalization to the Butterfly theorem also with the help of the Geometer's Sketchpad had compared the significance of his discovery to that of GLaD.

A less known, although no less simple, construction found by a college student, Nathan Besteman, has been reported in the recent issue of *Mathematics Teacher*.

What if applet does not run? |

Let AB be a given segment and C a point on the line AB as in the above diagram. Let D and E be the intersections of two circles: one centered at A, the other at C. By the symmetry of the construction, DE is perpendicular to AC. Denote the point of intersection of DE and AC as P and the midpoint of DA as Q. The right triangles APD and ACQ that share an angle at A are similar. Wherefrom we get,

(1) | AP/AQ = AD/AC |

However, AQ = AD/2 by construction, and AD = AB, as radii of the same circle. Thus (1) implies

(2) | AP = AB^{2}/(2·AC). |

Now, if C was chosen so that AC = N·(AB/2), then (2) yields

(3) | AP = AB/N, |

which leads to the following construction: find M, the midpoint of AB, and then C that satisfies AC = N·AM. Draw the circles with centers A and C and the radii AB and AC, respectively. These intersect at points D and E. DE meets AB at the point P such that AP = AB/N.

### References

- N. Besteman and J. Ferdinands,
__Another Way to Divide a Line Segment into__,*n*Equal Parts*Mathematics Teacher*, Vol. 98, No. 6 (Feb. 2005), pp. 428-433 - E. Maor,
*The Pythagorean Theorem*, Princeton University Press, 2007

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Copyright © 1996-2018 Alexander Bogomolny