# midpoint

The concept of midpoint of line segment (http://planetmath.org/Midpoint) 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 , take a new point $T$ for the origin by using the linear substitutions of the form

 $x\;:=\;x^{\prime}\!+\!a,\qquad y\;:=\;y^{\prime}\!+\!b\quad\mbox{etc.}$

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 (http://planetmath.org/BasicPolynomial) 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 (http://planetmath.org/QuadraticSurfaces)).

## References

• 1 Felix Iversen: Analyyttisen geometrian oppikirja. Tiedekirjasto Nr. 19.  Second edition.  Kustannusosakeyhtiö Otava, Helsinki (1963).
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