Inflection points of a real planar singular cubic curve

Solution 1:

Inflection points are 3-torsion points for the chord and tangent group law on the nonsingular points of a plane cubic. But the chord-and-tangent group law on the smooth real points of a real nodal plane cubic is isomorphic to $\Bbb R^\times$, which has only the identity as a 3-torsion point.

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