Segre's theorem

In projective geometry Segre's theorem, named after the Italian mathematician Beniamino Segre, is the statement:

This statement was assumed 1949 by the two Finnish mathematicians G. Järnefelt and P. Kustaanheimo and its proof was published in 1955 by B. Segre.

A finite pappian projective plane can be imagined as the projective closure of the real plane (by a line at infinity), where the real numbers are replaced by a finite field K. Odd order means that |K| = n is odd. An oval is a curve similar to a circle (see definition below): any line meets it in at most 2 points and through any point of it there is exactly one tangent. The standard examples are the nondegenerate projective conic sections.

For pappian projective planes of even order there are always ovals which are not conics. In an infinite plane there exist ovals, which are not conics. In the real plane one just glues a half of a circle and a suitable ellipse smoothly.

The proof of Segre's theorem, shown below, uses the 3-point version of Pascal's theorem and a property of a finite field of odd order, namely, that the product of all the nonzero elements equals -1.

If ${\displaystyle |g\cap {\mathfrak {o}}|=0}$ the line ${\displaystyle g}$ is an exterior (or passing) line; in case ${\displaystyle |g\cap {\mathfrak {o}}|=1}$ a tangent line and if ${\displaystyle |g\cap {\mathfrak {o}}|=2}$ the line is a secant line.

...
Wikipedia