Fraleigh(7th) Theorem 12.5: Every finite group $G$ of isometries of the plane is isomorphic to either $Z_n$ or to a dihedral group $D_n$ for some positive integer $n$. (Note: An isometry of $\mathbb{R}^2$ is a permutation of $\mathbb{R}^2$ that preserves distances.)
Proof: First we show that there is a fixed point by all of $G$. Let $G=\{f_1,\cdots,f_m\}$ and let $(x_i,y_i):=f_i(0,0)$. Then the point $P=(\bar{x},\bar{y})=\left( \dfrac{x_1+\cdots+x_m}{m},\dfrac{y_1+\cdots+y_m}{m}\right)$ is the centroid of the set $S=\{(x_i,y_i)\ |\ i=1,\cdots,m\}$. The isometries in $G$ permute the points in $S$ among themselves, since if $f_i f_j=f_k$ then $f_i(x_j,y_j)=f_if_j(0,0)=f_k(0,0)=(x_k,y_k)$. It can be shown that the centroid of a set of points is uniquely determined by its distances from the points, and since each isometry in $G$ just permutes the set $S$, it must leave the centroid $(\bar{x},\bar{y})$ fixed. Thus $G$ consists of the identity, rotations about $P$, and reflections across a line through $P$. ......
First, I can't understand the bold text. How can I change the phrase "the centroid of a set of points is uniquely determined by its distances from the points" into mathmatical words? And thus why the centroid is left fixed by all of $G$?
Second, What is the name of the above theorem? I saw some sites calling it Leonardo's theorem, but there is no search result of this name in wiki. Is there any other name?