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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