Question 12
Given a fixed regular pentagon $ABCDE$ with side $1$. Let $M$ be an arbitrary point inside or on it. Let the distance from $M$ to the closest vertex be $\mathbf{r}_1$, to the next closest be $\mathbf{r}_2$ and so on, so that the distances from $M$ to the five vertices satisfy $\mathbf{r}_1 \leq \mathbf{r}_2 \leq \mathbf{r}_3 \leq \mathbf{r}_4 \leq \mathbf{r}_5$. Find
a) the locus of $M$ which gives $\mathbf{r}_3$ the minimum possible value,
b) the locus of $M$ which gives $\mathbf{r}_3$ the maximum possible value.
a) the locus of $M$ which gives $\mathbf{r}_3$ the minimum possible value,
b) the locus of $M$ which gives $\mathbf{r}_3$ the maximum possible value.