Circle-Point Midpoint Theorem

Taking the locus of Midpoints from a fixed point to a circle of radius $r$ results in a circle of radius $r/2$. This follows trivially from

$$\begin{eqnarray*} {\bf r}(\theta)&=&\left[\begin{array}{c}-x\\ 0\end{array}\rig... ...\over 2}}x\\ {\textstyle{1\over 2}}\sin\theta\end{array}\right]. \end{eqnarray*}$$

References

Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, p. 17, 1929.