<s>[INST] How find minimum of $\sqrt{(a+7)^2+(a+10)^2}+\sqrt{(a-b)^2+(a+5)^2}+\sqrt{(c-a)^2+(c-3)^2}+\sqrt{(c-6)^2+(c-8)^2}$ let $a,b,c\in R$,then Find this mimimum of the value
$$\sqrt{(a+7)^2+(a+10)^2}+\sqrt{(a-b)^2+(a+5)^2}+\sqrt{(c-a)^2+(c-3)^2}+\sqrt{(c-6)^2+(c-8)^2}$$
My try: since
$$\sum_{i=1}^{n}\sqrt{a^2_{i}+b^2_{i}}=\sqrt{(\sum_{i=1}^{n}a_{i})^2+(\sum_{i=1}^{n}b_{i})^2}$$
But can't use full,Thank you
 [/INST] Hint:
An alternative to Minkowski - Consider a path through the points $(b, a), (a, -5), (-7, a+5), (c-10, c+5), (-4, 13)$.  The LHS is the length of this path, which obviously gets minimised when all these points are on a straight line, in order. Can you find $a, b, c$ so that these are on a straight line?
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