X = R^2 d(x,y) = (x1 - y1)^2 + (x2 - y2)^2
Having trouble proving the 3 axioms...
The first two axioms are trivial, so you are left to prove:
$\displaystyle d(x,y)+d(y,z) \ge d(x,z)$
but $\displaystyle d(a,b)=[\rho(a,b)]^2$ where $\displaystyle \rho$ is the usual metric on $\displaystyle \mathbb{R}^2$
So:
$\displaystyle d(x,y)+d(y,z) = [\rho(x,y)]^2+[\rho(y,z)]^2=[\rho(x,y)+\rho(y,z)]^2-2\rho(x,y)\rho(y,z)$ $\displaystyle \ge [\rho(x,y)+\rho(y,z)]^2$
....................... $\displaystyle \ge [\rho(x,z)]^2=d(x,z)$
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