What is the minimum of $\left|z-2(1+i)\right|+\left|z+1-5i\right|+\left|z-6+2i\right|$ over all complex numbers?

Question

Find the Least value of $\left|z-2(1+i)\right|+\left|z+1-5i\right|+\left|z-6+2i\right|$

My try:: Let $A(2,2)$ and $B(-1,5)$ and $C(6,-2)$ and $P(x,y)$ be a point

Here $A,B$ and $C$ are the point of a $\triangle$ and $P(x,y)$ be any point inside $\triangle ABC$

So We have to minimize $PA+PB+PC$

How Can I Minimize it, plz explain me in Detail

and also what should be the answer of that Question,

Thanks

Look at your three points a little more closely. One of them is wrong. (And with the right point, it becomes much easier.) — Daniel Fischer, Jul 22 '13 at 17:25
Now I have edited it, To lab bhattacharjee would you like to explain me in detail, Thanks — juantheron, Jul 22 '13 at 17:36
Please find : http://stackoverflow.com/questions/4609846/find-point-which-sum-of-distances-to-set-of-other-points-is-minimal — lab bhattacharjee, Jul 22 '13 at 17:37
But It is Given for Sum of Square of its Distance like for $PA^2+PB^2+PC^2$ — juantheron, Jul 22 '13 at 17:55
Related : http://math.stackexchange.com/questions/44386/find-the-point-in-a-triangle-minimizing-the-sum-of-distances-to-the-vertices and http://math.stackexchange.com/questions/58868/minimizing-sum-of-distances — lab bhattacharjee, Jul 22 '13 at 18:18

score 3 · Accepted Answer · answered Jul 22 '13 at 17:40

3

Hint:

Notice that your three points are collinear, so that the point you're looking for lies on the line defined by the three points.

By drawing a picture, and testing out some different points on the line, see if you can figure out which point on the line is your desired point.

answered Jul 22 '13 at 17:40

Jared

Got it, Now How can I Calculate That perticular point, Thanks – juantheron Jul 22 '13 at 17:52
Now If we have a Question like , Let $A(2,2)$ and $B(-1,5)$ and $C(6,2)$ and $P(x,y)$ be a point
Here $A,B$ and $C$ are the point of a $\triangle$ and $P(x,y)$ be any point inside $\triangle ABC$

So We have to minimize $PA+PB+PC$

How Can I Minimize it, plz explain me in Detail
– juantheron Jul 22 '13 at 17:54
1

@juantheron See http://en.wikipedia.org/wiki/Fermat_point. – Jaehyeon Seo Aug 04 '14 at 03:26

1 Answers1