Proving that a metric space with 3 points can be embedded isometrically into $\mathbb{R}^2$ My definition of an isometric embedding is that if $(M_2,d_1)$ and $(M_2,d_2)$ are metric spaces, then $G:M_1 \to M_2$ is an isometric embedding if $d_2(G(x),G(y)) = d_1(x,y)$ for all $x,y \in M_1$. 
I would like to show that any metric space that has 3 points can be embedded isometrically into $\mathbb{R}^2$ with the euclidean metric. My strategy has been to define the maps point by point, but it always seems that whatever first two points' map I define, I can never get the third to work. Does anyone know of a proof?
 A: Let's write $\{x_0, x_1, x_2\}$ as your 3-point metric space. Let's say that $y_0 = d(x_0, x_1)$, $y_1 = d(x_0, x_2)$, and $y_2 = d(x_1, x_2)$. We want to define an isometric embedding of this space into $\Bbb R^2$. Because I'm lazy, let's say $f(x_0) = 0$, and $f(x_1)=(0,y_0)$. Everything works so far: $d(f(x_0),f(x_1))=y_0=d(x_0,x_1)$. 
Now we want $d(f(x_0),f(x_2))=y_1$; so let's draw a circle of radius $y_1$ around $0$. If we can isometrically embed the 3-point metric space, $f(x_2)$ will have to lie on that circle. Similarly, we want $d(f(x_1),f(x_2))=y_2$; so let's draw a circle of radius $y_2$ around $(0,y_0)$. $f(x_2)$ will have to lie on this circle as well. So it suffices to find where our two circles intersect; if they intersect at all, then have $f(x_2)$ be one of the points of intersection. This is an isometric embedding.
So we need to check that the two circles intersect. But because $y_0+y_1 \geq y_2$ by the triangle inequality, this must be true (why?)
A: Let $X=\{A,B,C\}$ be the metric spaces with $3$ points and $a=d(B,C)$, $b=d(A,C)$ and $c=d(A,B)$. 
Then pick any points $D$, $E$ and $F$ in ${\mathbb R}^2$ such that $|EF|=a$, $|DF|=b$ and $|DE|=c$ (where $|PQ|$ denotes the Euclidean distance between two points $P$ and $Q$). It exists since $a+b\leq c$, $a+c\leq b$ and $b+c\leq a$. Then the application $f$ from $X$ into ${\mathbb R}^2$ defined by $f(A)=D$, $f(B)=E$ and $f(C)=F$ is clearly an isometry.
