1. **Problem Statement:** Given triangle $ABC$ with midpoints $D$, $E$, and $F$ of sides $BC$, $CA$, and $AB$ respectively, prove that the medial triangle $DEF$ is similar to $ABC$ with similarity ratio $\frac{1}{2}$ and that the midpoints lie on a circle with radius $\frac{R}{2}$ where $R$ is the circumradius of $ABC$. 2. **Similarity of Medial Triangle:** The medial triangle $DEF$ is formed by connecting midpoints of the sides of $ABC$. By the Midpoint Theorem, each side of $DEF$ is parallel to a side of $ABC$ and half its length. Therefore, $\triangle DEF \sim \triangle ABC$ with similarity ratio $\frac{1}{2}$. 3. **Distance from Midpoints to Circumcenter:** Let $O$ be the circumcenter of $\triangle ABC$. Since $D$, $E$, and $F$ are midpoints, each lies halfway between a vertex and the opposite vertex on the circumcircle. 4. **Radius of Circle through Midpoints:** The distance from $O$ to each midpoint is half the radius of the circumcircle: $$ OD = OE = OF = \frac{R}{2} $$ This means $D$, $E$, and $F$ lie on a circle centered at $O$ with radius $\frac{R}{2}$. **Final answer:** The medial triangle $DEF$ is similar to $ABC$ with ratio $\frac{1}{2}$, and the midpoints lie on a circle of radius $\frac{R}{2}$ centered at the circumcenter $O$.