We can use NMinimize to numerically search for the minimum:
In[]:=
NMinimize[Sqrt[x^2+y^2]+2*Sqrt[x^2+y^2-2*x+1]+Sqrt[x^2+y^2-6*x-8*y+25],{x,y}]
Out[]=
{5.47214,{x1.,y5.58782×
-9
10
}}
We can estimate the exact solution from the NMinimize result:
In[]:=
RootApproximant[5.4721359667669525`]
Out[]=
1+2
5
Here is a plot of the equation:
In[]:=
Plot3D[Sqrt[x^2+y^2]+2*Sqrt[x^2+y^2-2*x+1]+Sqrt[x^2+y^2-6*x-8*y+25],{x,-100,100},{y,-100,100}]
Out[]=
Symbolically minimizing takes too long for some reason...
In[]:=
Minimize[Sqrt[x^2+y^2]+2*Sqrt[x^2+y^2-2*x+1]+Sqrt[x^2+y^2-6*x-8*y+25],{x,y}]
Out[]=
$Aborted