Fermat point
The Fermat point (also called the Torricelli point) of a triangle 
is a point 
which has the minimum total distance to three vertices (i.e., 
).
![[asy] pair A=(2,4), B=(1,1), C=(6,1); pair pAB=rotate(60,B)*A, pCA=rotate(60,A)*C; path PC=pAB--C, PB=pCA--B; D(MP("A",A,2N)--MP("B",B,SW)--MP("C",C,SE)--cycle,green); D(C--pCA--A--pAB--B,red+dashed); DPA(PC^^PB,lightblue); D(MP("C'",pAB,NW),orange); D(MP("B'",pCA,NE),orange); D(A); D(B); D(C); D(MP("P",IP(PC,PB),E),blue); [/asy]](http://latex.artofproblemsolving.com/f/f/8/ff83e9cf8c39bc33ebf22d8365d8d13a2e7af16c.png)
Construction
A method to find the point is to construct three equilateral triangles out of the three sides from , then connect each new vertex to each opposite vertex, as these three lines will concur at first Fermat point.
See Also
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