Theorem (FermatTorricelli). Let all angles of a triangle be at most . Then the inner point of the triangle which makes the sum as little as possible, is the point from which the angle of view of every side is .
Proof. Let’s perform the rotation of about the point . When is the image of the point , the triangle is equilateral and its angles are . Let be any inner point of the triangle and its image in the rotation. We infer that if the sides of the triangle are all seen from in the angle , then the points , , , lie on the same line.
Remark. The point is called the Fermat point of the triangle .
- 1 Tero Harju: Geometria. Lyhyt kurssi. Matematiikan laitos. Turun yliopisto, Turku (2007).
|Date of creation||2013-03-22 19:36:39|
|Last modified on||2013-03-22 19:36:39|
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