Difference between revisions of "Fermat point"
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The '''Fermat point''' (also called the Torricelli point) of a triangle <math>\triangle ABC</math> is a point <math>P</math> which has the minimum total distance to three [[vertices]] (i.e., <math>AP+BP+CP</math>). | The '''Fermat point''' (also called the Torricelli point) of a triangle <math>\triangle ABC</math> is a point <math>P</math> which has the minimum total distance to three [[vertices]] (i.e., <math>AP+BP+CP</math>). | ||
| + | |||
| + | <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> | ||
==Construction== | ==Construction== | ||
Revision as of 05:02, 26 September 2014
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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