Sectrix of Maclaurin - Sectrix Properties

Sectrix Properties

Let where and are integers in lowest terms and assume is constructible with compass and straightedge. (The value of is usually 0 in practice so this is not normally an issue.) Let be a given angle and suppose that the sectrix of Maclaurin has been drawn with poles and according to the construction above. Construct a ray from at angle and let be the point of intersection of the ray and the sectrix and draw . If is the angle of this line then

so . By repeatedly subtracting and from each other as in the Euclidean algorithm, the angle can be constructed. Thus, the curve is an m-sectrix, meaning that with the aid of the curve an arbitrary angle can be divided by any integer. This is a generalization of the concept of a trisectrix and examples of these will be found below.

Now draw a ray with angle from and be the point of intersection of this ray with the curve. The angle of is

and subtracting gives an angle of


Applying the Euclidean Algorithm again gives an angle of showing that the curve is also an n-sectrix.

Finally, draw a ray from with angle and a ray from with angle, and let be the point of intersection. This point is on the perpendicular bisector of so there is a circle with center containing and . so any point on the circle forms an angle of between and . (This is, in fact, one of the Apollonian circles of P and P'.) Let be the point intersection of this circle and the curve. Then so


Applying a Euclidean algorithm a third time gives an angle of, showing that the curve is an (mn)-sectrix as well.

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