n-Sectors of the Angles of a Triangle

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The -sectors of an angle are the lines that divide the angle into equal parts. In a triangle, the bisectors are collinear, the trisectors lead to Morley's theorem, …. More modestly, this Demonstration provides a way to count the intersection points of the -sectors of the angles of a triangle as a function of , which is for a scalene triangle.

Contributed by: Jacques Marchandise (May 2011)
Open content licensed under CC BY-NC-SA



Let , , and be the sets of -sectors of the angles , , and , respectively. The number of points of intersection between any two of the sets is , hence, the number of points is at most . For example, the maximum number of points for is 3: one is the intersection of the bisectors of and , one is the intersection of the bisectors of and , and one is the intersection of the bisectors of and , but we know these three points coincide. The principle of inclusion-exclusion gives the number of distinct points:


When , this gives .

Now we want to count the distinct points for any value of . Some experimentation with this Demonstration shows that the result depends on the triangle's shape, especially its central or lateral symmetry. For example, for , the number of lines is nine, which is odd, so that the bisectors of each angle are in each of , , and . The number is 37 for an equilateral triangle with central symmetry, one for a scalene triangle with no symmetry, and nine for a isosceles with lateral symmetry; for , both are zero, and so on.

By tabulating the numbers for each set and applying Mathematica's built-in function FindSequenceFunction on the results, it appears (this is not a proof) that for a scalene triangle . For an isosceles triangle without a right angle the formula appears to be ; for an equilateral or right isosceles triangle the formulas are complicated.

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