Search

1. The resolvent, VI= x0 +.EX, + . x1 -I2x2 +* + E1-x 1, where E is a primitive n-th root of uiity, was introduced by Lagrange 2 in his memoirs devoted to the fundamental principles of the solutions of the cubic and quartic equations. Its entrance, however, into the field of geometry is very recent. If we represent any point P in the plane by the single complex number p, and if Mk (k 0, 1, * , n 1) represents the n vertices of a positivelyordered polygon, then when the co6rdinates pA of these vertices are subject to one and only one condition, namely that

In a paper,* which discussed the imprimitive group of ordelr 5184 in S3, the existence of an imprimitive group of order 216 in S2 connected with the Hesse configurations was indicated. The purpose of this paper is to discuss this planar group and the geometry associated with it. Of special interest is a configuration of twelve triangles which possesses important properties. These are dealt with in Section II. In Section I is a study of four-fold perspective triangles. A new canonical form is used which leads to a ruler construction for them and to a compass construction for a Clebsch six-point. I. FOUR-FOLD PERSPECTIVE TRIANGLES.