Constant rations for inflection points of a cubic curve with a node or an acnode

In this paper, projective invariants of cubic curves with a node or an acnode are obtained. It is proved that on the projective plane every two inflection points of a cubic curve with a node (acnode) are in an equianharmonic ratio with the points of tangents of the given curve at its node (acnode) located on the line containing these inflection points. And every three inflection
points of such a curve are in a quasi-anharmonic ratio with the point on tangent of this curve at its node (acnode) located on the line containing these inflection points.
It is established that in the projective plane any two crunodal (acnodal) cubics are projectively equivalent.
It is proved that four lines containing the node (acnode) of a cubic, namely: the line of the inflection points, the tangent and pseudotangent to the curve at the inflection point, the tangent to the curve at the point conjugate to the inflection point, are in a constant cross ratio equal to −3. Based on this fact, a number of properties of cubic curves with a node (acnode) in the Euclidean plane 𝐸2 are substantiated. Let us present some of the proved properties, denoting the cubic curve by the symbol 𝜎, and its node or acnode by the symbol 𝐼.

1. If the tangents of 𝜎 at an acnode 𝐼 pass through circle points of the plane 𝐸2, then the angle between any two lines, each of which connects the point 𝐼 with the inflection point of this curve, is equal to 𝜋/3.

2. The pseudotangent at the point 𝐼 divides the strip between the parallel tangents of 𝜎 passing through 𝐼, in the ratio of three to one, counting from the tangent of 𝜎 at the conjugate point with 𝐼, if and only if, when the line of the inflection points of 𝜎 coincides with the absolute line of the plane 𝐸2.

3. The tangent of 𝜎 at the conjugate point with 𝐼 divides the strip between the mutually parallel pseudotangents at the point 𝐼 and the line of the inflection points of 𝜎 in the ratio of three to one, counting from the pseudotangent, if and only if, when the tangent line of 𝜎 at the point 𝐼 coincides with the absolute line of the plane 𝐸2.

4. The line of the inflection points of the curve 𝜎 divides the strip between the parallel tangents to 𝜎 passing through 𝐼 in the ratio of three to one, counting from the tangent line of 𝜎 at the point 𝐼, if and only if when the pseudotangent of the curve 𝜎 at the point 𝐼 coincides with the absolute line of the plane 𝐸2.

5. The tangent of the curve 𝜎 at the point 𝐼 divides the strip between the line of inflection points and the parallel to it pseudotangent of 𝜎 at 𝐼 in the ratio of three to one, counting from the line inflection points, if and only if the tangent of the curve 𝜎 at the conjugate point with 𝐼 coincides with the absolute line of the plane 𝐸2.

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