Bitangents of Algebraic Curves
In general, an algebraic curve will have infinitely many secant lines, but only finitely many bitangents.
Bézout's theorem implies that a plane curve with a bitangent must have degree at least 4. The case of the 28 bitangents of a quartic was a celebrated piece of geometry of the nineteenth century, a relationship being shown to the 27 lines on the cubic surface.
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