Fermat point
The two Fermat points, named after the French judge and mathematician Pierre de Fermat, are among the special points of a triangle. About the sides of a given triangle ABC to draw three equilateral triangles. Combining the newly added points A1, B1 and C1 with the opposite corners of the triangle (ie A, B or C), then these links intersect at a point F. This is called the first Fermat point of the triangle. The first Fermat point is in the business mathematics, specifically in the site planning application. Suppose three companies plan to build a central warehouse such that the transport costs for this central warehouse are minimal. The central warehouse would be built on the site of Fermatpunkts, if one imagines the position of the three companies as a triangle because the sum of the distances to the corners of the triangle is minimal for the Fermat point (where all the angles in a triangle must be less than 120 ° ).
The second Fermat point of a triangle is given by the same construction as the first Fermat point, only you have the equilateral triangles each not build " out" on the sides of the triangle, but "inwardly". It has essentially the same characteristics as the first Fermat point, however, appear with him always one side at an angle of 120 ° and two sides at an angle of 60 °.
Properties
- If all the angles of triangle ABC are less than 120 °, then the first Fermat point of the triangle is the point inside the triangle, appear from which all three sides at an angle of 120 °; this means.
- If an angle of triangle ABC is greater than or equal to 120 ° ( for example ), then instead and.
This case is always for the second Fermat point.
- All of the given angle triangle ABC is smaller than 120 °, the first Fermat point is that point for which the sum of the distances from the corners of the triangle ABC (the sum ) assumes the smallest possible value.
The proof of this fact comes from the Italian Evangelista Torricelli. Wherefore they say, sometimes from the Fermat - Torricelli point. If, however, an angle of the triangle ABC is greater than or equal to 120 °, then no longer is the first Fermat point is the point with the smallest sum of the distances, but the corner at which the angle lies.
- The two Fermat points are isogonal conjugated to the two isodynamic points.
- The Fermat points lie on the Kiepert hyperbola.
Evidence
We use the properties of vectors and their scalar product in the Euclidean plane.
If all interior angles are less than 120 ° in the triangle ABC, we can construct the Fermat point F inside the triangle ABC. Then we set
If F is the Fermat point, then applies so that we get the equation from Lemma 1 by definition.
From Lemma 2 we see that
From these three inequalities and the equation of Lemma 1 follows
This is true for any point X in the Euclidean plane. Thus we have shown that if X = F, then the value is minimal. Q.e.d.