[alogo] Yates trisector

Originally invented by Laisant and commented by Brocard, the trisector of Yates is a mechanical device (linkage) to exactly trisect an angle <(AOB). The device uses a rhombus OBCD of fixed side-length a (OB = OD = DC = BC = DA = HE), whose side OB is fixed and vertex C movable. DE is on the extension of OD and has length DE = b. EG = DF is taken to be of length c, such that b2 = a*c. This guarantees that DE bisects angle ADC (use circle through HGD and the symmetry of linkage EGFD).
For all positions C lines OE and OC trisect angle AOB.

[0_0] [0_1] [0_2] [0_3]
[1_0] [1_1] [1_2] [1_3]

The connecting points of the linkage describe interesting curves, of which most easily studied is the limacon of Pascal described by point A, as C moves on the circle k(O, OC). This is seen by taking the symmetric P of B with respect to O. PA is orthogonal to AC, which is tangent to circle (k). Thus A desccribes the pedal of circle (k) with respect to the fixed point P. This is one of the well known generation kinds of the limacon.

See Also



Robert C. Yates A Trisector National Mathematics Magazine, Vol. 12, No. 7. (Apr., 1938), pp. 323-324.
H.Brocard Note sur un compas trisecteur propose' par M. Laisant Bulletin de la S.M.F. tome 3(1875) p. 47-48

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