To find the minimum value of $|z+1|+|z-1|+|z-i|$ where $z\in \Bbb C$.

To find the minimum value of $|z+1|+|z-1|+|z-i|$ where $z\in \Bbb C$. Options:

$(A) \ 2$

$(B) \ 2\sqrt2$

$(C) \ 1+\sqrt3$

$(D) \ \sqrt5$

My logic is the sum will be minimum iff $z\in \Bbb C$ is any one of the three fixed points $1,-1,i$. And by calculation we see that the sum is min when take $z=i$.Is the solution correct?

Know that its not a good solution to the problem....searching for an elegant one...Suggestion reqd..

One can apply Fermat-Torricelli point as given in solution below by Quang Hoang and it is a good solution to the problem geometrically.....but this can be applied only if I know the result...searching a solution of this from known basic results of analysis...

Hint: Look at the triangle with three vertices: $1$, $-1$, and $i$ on the complex plane. The answer to the question is the Fermat-Torricelli point.