Fermat point

In geometry, the first Fermat point, or simply the Fermat point, also called Torricelli point, is the solution to the problem of finding a point F inside a triangle ABC such that the total distance from the three vertices to point F is the minimum possible. It is so named because this problem is first raised by Fermat in a private letter.

Construction

To locate the Fermat point of a triangle with largest angle at most 120°:

#Construct three regular triangles out of the three sides of the given triangle.
#For each new vertex of the regular triangle, draw a line from it to the opposite triangle's vertex.
#These three lines intersect at the Fermat point.

The reader should be careful not to confuse the Fermat point with the first isogonic center also known as "X"(13).Entry X(13) in the [http://faculty.evansville.edu/ck6/encyclopedia/ETC.html Encyclopedia of Triangle Centers] ] Whilst the two points are one and the same provided no angle of the triangle exceeds 120° the coincidence ends there. When a triangle has an angle greater than 120° the first isogonic center always lies outside the triangle whereas the Fermat point is sited at the obtuse angled vertex. This means that in such a triangle the two points are never the same.

Geometry

Given any Euclidean triangle ABC and an arbitrary point P let d(P) = PA+PB+PC. The aim of this section is to identify a point P₀ such that d(P₀) < d(P) for all P ≠ P₀. If such a point exists then it will be the Fermat point. In what follows Δ will denote the points inside the triangle and will be taken to include its boundary Ω.

A key result that will be used is the dogleg rule which asserts that if a triangle and a polygon have one side in common and the rest of the triangle lies inside the polygon then the triangle has a shorter perimeter than the polygon.
[ If AB is the common side extend AC to cut the polygon at X. Then by the triangle inequality the polygon perimeter > AB+AX+XB = AB+AC+CX+XB ≥ AB+AC+BC. ]

Let P be any point outside Δ. Associate each vertex with its remote zone; that is, the half-plane beyond the (extended) opposite side. These 3 zones cover the entire plane except for Δ itself and P clearly lies in either one or two of them. If P is in two (say the B and C zones) then setting P' = A implies d(P') = d(A) < d(P) by the dogleg rule. Alternatively if P is in only one zone, say the A-zone, then d(P') < d(P) where P' is the intersection of AP and BC. So for every point P outside Δ there exists a point P' in Ω such that d(P') < d(P).

Case 1. The triangle has an angle ≥ 120^o.

Without loss of generality suppose that the angle at A is ≥ 120^o. Construct the equilateral triangle AFB and for any point P in Δ (except A itself) construct Q so that the triangle AQP is equilateral and has the orientation shown. Then the triangle ABP is a 60^o rotation of the triangle AFQ about A so these two triangles are congruent and it follows that d(P) = CP+PQ+QF which is simply the length of the path CPQF. As P is constrained to lie within ABC, by the dogleg rule the length of this path exceeds AC+AF = d(A). Therefore d(A) < d(P) for all P є Δ, P ≠ A. Now allow P to range outside Δ. From above a point P' є Ω exists such that d(P') < d(P) and as d(A) ≤ d (P') it follows that d(A) < d(P) for all P outside Δ. Thus d(A) < d(P) for all P ≠ A which means that A is the Fermat point of Δ. In other words the Fermat point lies at the obtuse angled vertex.

Case 2. The triangle has no angle ≥ 120^o.

Let P be any point inside Δ and construct the equilateral triangle CPQ. Then CQD is a 60^o rotation of CPB about C so d(P) = PA+PB+PC = AP+PQ+QD which is simply the length of the path APQD. Let P₀ be the point where AD and CF intersect. This point is commonly called the first isogonic center. By the angular restriction P₀ lies inside Δ moreover BCF is a 60^o rotation of BDA about B so Q₀ must lie somewhere on AD. Since CDB = 60^o it follows that Q₀ lies between P₀ and D which means AP₀Q₀D is a straight line so d(P₀) = AD. Moreover if P ≠ P₀ then either P₀ or Q₀ won't lie on AD which means d(P₀) = AD < d(P). Now allow P to range outside Δ. From above a point P' є Ω exists such that d(P') < d(P) and as d(P₀) ≤ d(P') it follows that d(P₀) < d(P) for all P outside Δ. That means P₀ is the Fermat point of Δ. In other words the Fermat point is coincident with the first isogonic center.

Derivation

Since the time the problem first appeared, many methods to arrive at the solution have been developed. One method is to simply rotate BEC, where E is an arbitrary point, 60º counter-clockwise. Now the distance to minimize is the same as the path AEE'C'. Obviously the solution is when it is a straight line, from which the construction method can be derived.

Proof

This proof will show that the three lines are concurrent. One proof, using properties of concyclic points, is as follows:

Suppose RC and BQ intersect at F, and two lines, AF and AP, are drawn. We aim to prove that AFP is a straight line.

Because AR = AB and AC = AQ by construction,

: $angle\; RAC\; =\; angle\; RAB\; +\; angle\; BAC$

: $angle\; BAQ\; =\; angle\; BAC\; +\; angle\; CAQ$

Since $angle\; RAB$ and $angle\; CAQ$ equal 60º, which are interior angles of an equilateral triangle, $angle\; RAC\; =\; angle\; BAQ$. This implies that triangles RAC and BAQ are congruent. Hence $angle\; ARF\; =\; angle\; ABF$ and $angle\; AQF\; =\; angle\; ACF$. By converse of angle in the same segment, ARBF and AFCQ are both concyclic.

Thus $angle\; AFB\; =\; angle\; AFC\; =\; angle\; BFC\; =\; 120$º. Because $angle\; BFC$ and $angle\; BPC$ add up to 180º, BPCF is also concyclic. Hence $angle\; BFP\; =\; angle\; BCP\; =\; 60$º. Because $angle\; BFP\; +\; angle\; BFA\; =\; 180$º, AFP is a straight line.

Q.E.D.

Another proof

Another approach to find a point within the triangle, from where sum of the distances to the vertices of triangle is minimum, is to use one of the optimization (mathematics) methods. In particular, method of the lagrange multipliers and the law of cosines.

We draw lines from the point within the triangle to its vertices and call them X, Y and Z. Also, let the lengths of these lines be x, y, and z, respectively. Let the acute angle between X and Y be α, Y and Z be β. Then the angle between X and Z is (2π − α − β). Using the method of lagrange multipiers we have to find the minimum of the lagrangian, which is expressed as:

: "x" + "y" + "z" + : "λ"₁ ("x"² + "y"² − 2"xy" cos("α") − "a"²) + : "λ"₂ ("y"² + "z"² − 2"yz" cos(β) − "b"²) + : "λ"₃ ("z"² + "x"² − 2"zx" cos("α" + "β") − "c"²) .

where "a", "b" and "c" are the lengths of the sides of the triangle.

Calculating the partial derivatives δ/δx, δ/δy, δ/δz, δ/δα, δ/δβ gives a system of 5 equations:

: δ/δx: 1 + λ₁(2x − 2y cos(α)) + λ₃(2x − 2z cos(α + β)) = 0

: δ/δy: 1 + λ₁(2y − 2x cos(α)) + λ₂(2y − 2z cos(β)) = 0

: δ/δz: 1 + λ₂(2z − 2y cos(β)) + λ₃(2z − 2x cos(α + β)) = 0

: δ/δα: λ₁y sin(α) + λ₃z sin(α + β) = 0

: δ/δβ: λ₂y sin(β) + λ₃x sin(α + β) = 0

After some algebraic manipulations equations for α and β separate from the rest of the parameters, giving:

: sin(α) = sin(β)

: sin(α + β) = −sin(β)

that gives: α = β = 120^o

Q.E.D.

Note: if one of the vertices of triangle has angle not less than 120^o, than Fermat point is at that vertex.

X(13) properties

* In case the largest angle of the triangle is not larger than 120º, the point minimizes the total distance from the three vertices to this point.
* The internal angles brought about by this point, that is, $angle\; AFB$, $angle\; BFC$, and $angle\; CFA$, are all equal to 120º.
* The circumcircles of the three regular triangles in the construction intersect at this point.
* Trilinear coordinates for the 1st isogonic center, "X"(13)::csc("A" + π/3) : csc("B" + π/3) : csc("C" + π/3), or, equivalently,:sec("A" − π/6) : sec("B" − π/6) : sec("C" − π/6).
* Trilinear coordinates for the 2nd isogonic center, "X"(14)::csc("A" − π/3) : csc("B" − π/3) : csc("C" − π/3), or, equivalently,:sec("A" + π/6) : sec("B" + π/6) : sec("C" + π/6). [Entry X(14) in the [http://faculty.evansville.edu/ck6/encyclopedia/ETC.html Encyclopedia of Triangle Centers] ]
* The isogonal conjugate of the 1st isogonic center is the 1st isodynamic point, "X"(15)::sin("A" + π/3) : sin("B" + π/3) : sin("C" + π/3). [Entry X(15) in the [http://faculty.evansville.edu/ck6/encyclopedia/ETC.html Encyclopedia of Triangle Centers] ]
* The isogonal conjugate of the 2nd isogonic center is the 2nd isodynamic point, "X"(16)::sin("A" − π/3) : sin("B" − π/3) : sin("C" − π/3). [Entry X(16) in the [http://faculty.evansville.edu/ck6/encyclopedia/ETC.html Encyclopedia of Triangle Centers] ]
* The following triangles are equilateral::antipedal triangle of "X"(13):antipedal triangle of "X"(14):pedal triangle of the "X"(15):pedal triangle of the "X"(16):circumcevian triangle of "X"(15):circumcevian triangle of "X"(16)
* The lines "X"(13)"X"(15) and "X"(14)"X"(16) are parallel to the Euler line. The three lines meet at the Euler infinity point, "X"(30).
* The 1st isogonic center, 2nd isogonic center, circumcenter, nine-point center lie on a Lester circle.

History

This question was proposed by Fermat, as a challenge to Evangelista Torricelli. He solved the problem in a similar way to Fermat's, albeit using intersection of the circumcircles of the three regular triangles instead. His pupil, Viviani, published the solution in 1659. [MathWorld|urlname=FermatPoints |title=Fermat Points]

ee also

*Geometric median or Fermat–Weber point, the point minimizing the sum of distances to more than three given points.
*Lester's theorem

References

External links

*" [http://demonstrations.wolfram.com/FermatPoint/ Fermat Point] " by Chris Boucher, The Wolfram Demonstrations Project.