I’m trying to show that $d(x,y)= |x_1-x_2|^2 + |y_1-y_2|^2$ is a metric on $\mathbb R^2$ where $x=(x_1,y_1)$ and $y=(x_2,y_2)$
I have stuck on triangle inequality.
I must show $d(x,y) \leq d(x,z) + d(z,y)$ namely $|x_1-x_2|^2 + |y_1-y_2|^2 \leq |x_1-x_3|^2 + |y_1-y_3|^2 + |x_3-x_2|^2 + |y_3-y_2|^ 2$
I have tried to use $|x_1-x_2|=|x_1-x_3+x_3-x_2| \leq |x_1-x_3| + |x_3-x_2| $ but I couldn’t obtain the triangle inequality. I could get $|x_1-x_2|^2 + |y_1-y_2|^2 \leq 2|x_1-x_3|^2 + 2|y_1-y_3|^2 + 2|x_3-x_2|^2 + 2|y_3-y_2|^ 2$ via using $2ab \leq a^2 + b^2$ inequality.
I know it is probably a piece of cake but I cannot see it. I need some hints.
Thanks in advance for any help.