Consider $X \times Y$ for metric spaces $X$ and $Y.$
I'm trying to show that $d_E(p,q) = \sqrt{d_X(p,q)^2 + d_Y(p,q)^2}$ fulfills the triangle inequality, but I can't workout the algebra so that things fall into place and it is clear that $$d_E(p_1,p_3) \leq d_E(p_1,p_2) + d_E(p_2,p_3).$$
I avoided writing out everything that I have written on paper because it is messy, but here is a little bit:
Consider $$d_E(p,p'') = \sqrt{d_X(p,p'')^2 + d_Y(p,p'')^2}$$ and $$d_E(p,p') = \sqrt{d_X(p,p')^2 + d_Y(p,p')^2}$$ and $$d_E(p',p'') = \sqrt{d_X(p',p'')^2 + d_Y(p',p'')^2}.$$ It must be shown that $$d_E(p,p'') \leq d_E(p,p') + d_E(p',p''),$$ or that $$d_E(p,p'')^2 \leq (d_E(p,p') + d_E(p',p''))^2.$$
I then proceeded to expand it out via the square, and it became messy and not, in the end, clear that the left was less than or equal to the right.
Note that $d_X(x_1,x_3)≤d_X(x_1,x_2)+d_X(x_2,x_3)$ and $d_Y(y_1,y_3)≤d_Y(y_1,y_2)+d_Y(y_2,y_3)$. Let us put $r_1=d_X(x_1,x_2),s_1=d_X(x_2,x_3),r_2=d_Y(y_1,y_2),s_2=d_Y(y_2,y_3)$.
Note that, $[(r_1+s_1)^2+(r_2+s_2)^2]^\frac{1}{2}≤(r_1^2+r_2^2)^\frac{1}{2}+(s_1^2+s_2^2)^\frac{1}{2}$(expand both sides to reduce shorter form and then apply cauchy-schwarz). Hence $d_E((x_1,y_1),(x_3,y_3))≤[(d_X(x_1,x_2))^2+(d_Y(y_1,y_2))^2]^\frac{1}{2} +[(d_X(x_2,x_3))^2+(d_Y(y_2,y_3))^2]^\frac{1}{2}=d_E((x_1,y_1),(x_2,y_2))+d_E((x_2,y_2),(x_3,y_3))$.