### On the Positions of the Imaginary Points of Inflexion and Critic Centers of a Real Cubic

The Annals of Mathematics, 2nd Ser., Vol. 23, No. 4 (June 1922), 287-291

Introduction

In the extensive study of the configuration formed by the points of inflexion of a real cubic, it appears that no one has considered the possible positions of the six imaginary points of the group when the real three points are fixed. This is worthy of consideration for these two sets of points are so related that, while the three collinear real points of inflexion impose only five conditions and hence determine a fourfold infinite system of cubics in a plane, not one of the six points can be chosen arbitrarily. The following gives a construction for such a set of six points when the three real points are taken arbitrarily on a line; and by a generalization accounts for all such possible sets of six points.

The construction for the six imaginary points of inflexion also serves to show the positions of the twelve critical centers for the non-singular real cubic.

Construction

Let any three real points I_{1}, I_{2}, I_{3} on an arbitrary real line be taken as points of inflexion for a real cubic. Let Σ be any other real point. Join Σ to the points I_{i} and construct the fourth harmonic to each one of these lines with respect to the other two. Denote the fourth harmonic to the line through I_{1} by h_{1} and similarly for I_{2} and I_{3}. Through any one of the three points, say I_{1}, draw an arbitrary real line intersecting h_{2} and h_{3} in V_{2} and V_{3}. Draw the lines I_{2}V_{3}, I_{3}V_{2} intersecting in V_{1} on h_{1}. The projections upon the sides of the triangle V_{1}, V_{2}, V_{3} through Σ, of the two points equianharmonic to the three points I, are six imaginary points which together with I_{i} form an inflexional group for a real cubic.