The concept of midpoint of line segment is a special case of the midpoint of a curve or arbitrary figure in $\mathbb{R}^2$ or $\mathbb{R}^3$ .

A point $T$ is a midpoint of the figure $f$ , if for each point $A$ of $f$ there is a point $B$ of $f$ such that $T$ is the midpoint of the line segment $AB$ . One says also that $f$ is symmetric about the point $T$ .

Given the equation of a curve in $\mathbb{R}^2$ or of a surface $f$ in $\mathbb{R}^3$ , one can, if necessary, take a new point $T$ for the origin by using the linear substitutions of the form $$x := x'\!+\!a, \quad y := y'\!+\!b \quad \mbox{etc.}$$ Thus one may test whether the origin is the midpoint of $f$ by checking whether $f$ always contains along with any point $(x,\,y,\,z)$ also the point $(-x,\,-y,\,-z)$ .

It is easily verified the

Theorem. If the origin is the midpoint of a quadratic curve or a quadratic surface, then its equation has no terms of degree 1.

Similarly one can verify the generalisation, that if the origin is the midpoint of an algebraic curve or surface of degree $n$ , the equation has no terms of degree $n\!-\!1$ , $n\!-\!3$ and so on.

Note. Some curves and surfaces have infinitely many midpoints (see quadratic surfaces).

Bibliography

1

FELIX IVERSEN: Analyyttisen geometrian oppikirja. Tiedekirjasto Nr. 19. Second edition. Kustannusosakeyhtiö Otava, Helsinki (1963).