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Angles and Trigonometry > Angle Sections

One of the basic constructions taught in high school geometry is the bisection of an angle. This construction splits an Euclidean angle into two equal angles. In 1837, Pierre Wantzel proved that no Euclidean construction trisects an arbitrary angle. In taxicab geometry, this problem is solved by the simplicity of taxicab angles - they are simply line segments. In fact, a construction exists to arbitrarily split a taxicab angle into any number of equal angles.

Arbitrary sectioning of taxicab angles was first mentioned by a referee of an early version of the Thompson-Dray paper on taxicab angles and trigonometry. It was later examined by Robert Dawson in an article on axiomatic taxicab geometry. Through all of this, it was never precisely clear whether arbitrary sectioning of a taxicab angle could be done with only taxicab constructs. Many Euclidean constructs exist to arbitrarily split a line segment, so Euclidean constructs could be used to arbitrarily section a taxicab angle. What would this construction look like with only taxicab constructs?

CONSTRUCTION: To split a line segment with slope 1 into n ≥ 3 equal parts using taxicab constructions

 Given a line segment AB of taxicab length 2l with slope 1 with A at the origin, construct a taxicab circle of radius 2l with center at B (see Figure 1). Likewise construct a taxicab circle of radius 2l with center at A. If n > 3, at the intersection of the circle around A with the extension of the line segment, construct a circle of radius 2l. If n > 4, repeat the last step n - 4 additional times. Construct a line from the bottom corner of the last circle to point B creating an intersection P with the circle about B. Construct a line from P to the top of the circle about A creating an intersection C with the line segment AB. The taxicab length of the line segment AC is 2l / n.

FIGURE 1: Construction to split a line segment into n equal parts (n = 3 and n = 4 shown).

Proof: Let A = (0,0) and B = (l,l). The bottom corner of the last constructed circle is located at ((3 - n)l,(1-n)l) so the line from such a point to B is

This line intersects the line y = -x at point P with x-coordinate x = l / (n-1). The line from this intersection to the point (0,2l) at the top of the circle centered at A is

This line intersects the line y = x at x = l / n giving a taxicab distance from A to this point C of

The presented construction need not be limited to lines of slope 1. This was merely done for the simplicity of calculation and because t-radian angles are defined along diagonal lines. Figure 2 illustrates the general construction.

FIGURE 2: Generalized construction to split a line segment into n equal parts (n = 3 and n = 4 shown).

With the general construction, we can arbitrarily section taxicab angles since a t-radian is merely the length along a line of slope -1. And, not only have we developed a method to arbitrarily section taxicab angles and line segments, but we have also developed yet another method for arbitrarily splitting Euclidean line segments!

References
[1] Dawson, Robert J. MacG. Crackpot Angle Bisectors! Mathematics Magazine, Vol. 80, No. 1 (Feb 2007), pp. 59-64.
[2] Thompson, Kevin. Arbitrary Sectioning of Angles in Taxicab Geometry (unpublished).
[3] Thompson, Kevin and Tevian Dray. Taxicab Angles and Trigonometry, The Pi Mu Epsilon Journal, Vol. 11, No. 2 (Spring 2000), pp. 87-96.
[4] Wantzel, Pierre L. Recherches sur les moyens de reconnaître si un Problème de Géométrie peut se résoudre avec la règle et le compas, Journal de Mathématiques Pures et Appliquées, Vol. 1, No. 2 (1837), pp. 366-372.

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