Problem of the Month (January 1999)

If we have n distinct points in the plane, they determine

triangles, and therefore 3

angles. Each of these angles θ satisfies 0 ≤ θ ≤ π, with equality if the points are colinear. How often can an angle θ occur?

Let A(n,θ) be the maximum number of times angle θ can be formed by n points. For example, the values of A(3,θ) are given in the table below:

Angles Formed by Three Points

θ	A(3,θ)
[ 0 , π/2 )	2
[ π/2 , π ]	1
π/3	3

That is, angles θ a right angle or larger can be formed only once, and angles θ smaller than a right angle can be formed twice, except for θ=π/3 which can be formed 3 times.

What are the values of A(4,θ)? How about A(n,θ) for larger n? Can you prove your answers?

ANSWERS

This month's results are temporarily unavailable, because I have a student doing his senior research project on this topic. The results will return in 2009. Thank you for your patience.

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 1/28/99.