# Bisecting Arcs

M. Klamkin tells the following story:

The next problem is a very nice one in Polya [2, p. 186]; I managed to catch D.J. Newman on it as well as other mathematicians. A bisecting arc is one which bisects the area of a given region. First, I asked what is the shortest bisecting arc of a circle. Usually, the fast reply is that it is a diameter. Secondly, I asked what is the shortest bisecting arc of a square. Again, a usual fast reply is that it is an altitude through the center. Finally, I asked what is the shortest bisecting arc of an equilateral triangle. By this time, Newman had suspected that I was setting him up (and I was) and almost was going to say the angle bisector. But he hesitated and said let me consider a chord parallel to the base and since this turns out to be shorter than an angle bisector, he gave this as his answer. Unfortunately for him, the correct answer is different.

## References

1. M. S. Klamkin, Mathematical Creativity in Problem Solving, in In Eves' Circles, J. M. Anthony (ed.), MAA, 1994
2. G. Polya, Mathematics and Plausible Reasoning, v 1, Princeton University Press, 1954, p 272