In
Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...
, the Fermat point of a
triangle
A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...
, also called the Torricelli point or Fermat–Torricelli point, is a point such that the sum of the three distances from each of the three vertices of the triangle to the point is the smallest possible or, equivalently, the
geometric median
In geometry, the geometric median of a discrete point set in a Euclidean space is the point minimizing the sum of distances to the sample points. This generalizes the median, which has the property of minimizing the sum of distances or absolute ...
of the three vertices. It is so named because this problem was first raised by
Fermat
Pierre de Fermat (; ; 17 August 1601 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he is recognized for his d ...
in a private letter to
Evangelista Torricelli
Evangelista Torricelli ( ; ; 15 October 160825 October 1647) was an Italian people, Italian physicist and mathematician, and a student of Benedetto Castelli. He is best known for his invention of the barometer, but is also known for his advances i ...
, who solved it.
The Fermat point gives a solution to the
geometric median
In geometry, the geometric median of a discrete point set in a Euclidean space is the point minimizing the sum of distances to the sample points. This generalizes the median, which has the property of minimizing the sum of distances or absolute ...
and
Steiner tree problem
In combinatorial mathematics, the Steiner tree problem, or minimum Steiner tree problem, named after Jakob Steiner, is an umbrella term for a class of problems in combinatorial optimization. While Steiner tree problems may be formulated in a ...
s for three points.
Construction
The Fermat point of a triangle with largest angle at most 120° is simply its first isogonic center or X(13), which is constructed as follows:
# Construct an
equilateral triangle
An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the ...
on each of two arbitrarily chosen sides of the given triangle.
# Draw a line from each new
vertex to the opposite vertex of the original triangle.
# The two lines intersect at the Fermat point.
An alternative method is the following:
# On each of two arbitrarily chosen sides, construct an
isosceles triangle
In geometry, an isosceles triangle () is a triangle that has two Edge (geometry), sides of equal length and two angles of equal measure. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at le ...
, with base the side in question, 30-degree angles at the base, and the third vertex of each isosceles triangle lying outside the original triangle.
# For each isosceles triangle draw a circle, in each case with center on the new vertex of the isosceles triangle and with radius equal to each of the two new sides of that isosceles triangle.
# The intersection inside the original triangle between the two circles is the Fermat point.
When a triangle has an angle greater than 120°, the Fermat point is sited at the obtuse-angled vertex.
In what follows "Case 1" means the triangle has an angle exceeding 120°. "Case 2" means no angle of the triangle exceeds 120°.
Location of X(13)
Fig. 2 shows the equilateral triangles attached to the sides of the arbitrary triangle .
Here is a proof using properties of
concyclic points
In geometry, a set of points are said to be concyclic (or cocyclic) if they lie on a common circle. A polygon whose vertices are concyclic is called a cyclic polygon, and the circle is called its ''circumscribing circle'' or ''circumcircle'' ...
to show that the three lines in Fig 2 all intersect at the point and cut one another at angles of 60°.
The triangles are
congruent
Congruence may refer to:
Mathematics
* Congruence (geometry), being the same size and shape
* Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure
* In modu ...
because the second is a 60° rotation of the first about . Hence and . By the converse of the
inscribed angle theorem
In geometry, an inscribed angle is the angle formed in the interior of a circle when two chords intersect on the circle. It can also be defined as the angle subtended at a point on the circle by two given points on the circle.
Equivalently, an ...
applied to the segment , the points are
concyclic
In geometry, a set of points are said to be concyclic (or cocyclic) if they lie on a common circle. A polygon whose vertices are concyclic is called a cyclic polygon, and the circle is called its ''circumscribing circle'' or ''circumcircle''. ...
(they lie on a circle). Similarly, the points are concyclic.
, so , using the
inscribed angle theorem
In geometry, an inscribed angle is the angle formed in the interior of a circle when two chords intersect on the circle. It can also be defined as the angle subtended at a point on the circle by two given points on the circle.
Equivalently, an ...
. Similarly, .
So . Therefore, . Using the
inscribed angle theorem
In geometry, an inscribed angle is the angle formed in the interior of a circle when two chords intersect on the circle. It can also be defined as the angle subtended at a point on the circle by two given points on the circle.
Equivalently, an ...
, this implies that the points are concyclic. So, using the
inscribed angle theorem
In geometry, an inscribed angle is the angle formed in the interior of a circle when two chords intersect on the circle. It can also be defined as the angle subtended at a point on the circle by two given points on the circle.
Equivalently, an ...
applied to the segment , . Because , the point lies on the line segment . So, the lines are
concurrent (they intersect at a single point).
Q.E.D.
Q.E.D. or QED is an initialism of the List of Latin phrases (full), Latin phrase , meaning "that which was to be demonstrated". Literally, it states "what was to be shown". Traditionally, the abbreviation is placed at the end of Mathematical proof ...
This proof applies only in Case 2, since if , point lies inside the circumcircle of which switches the relative positions of and . However it is easily modified to cover Case 1. Then hence which means is concyclic so . Therefore, lies on .
The lines joining the centers of the circles in Fig. 2 are perpendicular to the line segments . For example, the line joining the center of the circle containing and the center of the circle containing , is perpendicular to the segment . So, the lines joining the centers of the circles also intersect at 60° angles. Therefore, the centers of the circles form an equilateral triangle. This is known as
Napoleon's Theorem
In geometry, Napoleon's theorem states that if equilateral triangles are constructed on the sides of any triangle, either all outward or all inward, the lines connecting the centres of those equilateral triangles themselves form an equilateral tr ...
.
Location of the Fermat point
Traditional geometry
Given any Euclidean triangle and an arbitrary point let
The aim of this section is to identify a point such that