**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 (*m*−*n*)-sectrix as well.

Read more about this topic: Sectrix Of Maclaurin

### Famous quotes containing the word properties:

“A drop of water has the *properties* of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”

—Ralph Waldo Emerson (1803–1882)