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.
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.
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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