If a triangle △ABC is given, the sum of the distances from the fermat point p of the triangle to the three vertices A, B and C of the triangle is smaller than the sum of the distances to other points.
For each given triangle, there is only one special point.
1. If all three internal angles of the triangle are less than 120, then the three distance connecting lines just bisect the fillet where fermat point is located, that is, the opening angles of the three sides of the triangle are equal, which are all 120. So the fermat point of a triangle is also called the isocenter of a triangle.
(In Torricelli's solution, this point is described as follows: For each side of triangle ABC whose angle is less than 120, it is the bottom, then the regular triangle, and then the circumscribed circle of these three regular triangles. Torricelli pointed out that these three circumscribed circles will have * * * identical intersections, and this intersection is the necessary point. This is different from the special point of triangle known at that time. Therefore, this point is also called Torricelli point. )
2. If the internal angle of the triangle is greater than or equal to 120, the vertex of this obtuse angle is the point with the smallest distance sum.
Pierre de Fermat was a French lawyer and amateur mathematician in the17th century. Amateur is called amateur because Pierre de Fermat has a full-time job as a lawyer. According to the actual pronunciation of French and English, his surname is often translated into "Fermat" (note that it is the word "horse"). Fermat's Last Theorem is customarily called Fermat's Last Theorem in China. The original name of "last" in western mathematics field means that all other conjectures have been confirmed, and this is the last one.
The famous mathematical historian E. T. Bell called Pierre de Fermat "the king of amateur mathematicians" in his works in the early 20th century. Bell was convinced that Fermat was more successful than most professional mathematicians of Pierre de Fermat's contemporaries. However, Pierre de Fermat did not achieve any other achievements, and he gradually withdrew from people's field of vision. Considering that17th century is a century in which outstanding mathematicians are active, Bell thinks that Fermat is the most prolific star among mathematicians in17th century.
Fermat point problem was first put forward by French mathematician pierre de fermat in a letter to Italian mathematician Ivanjesta Torricelli (inventor of barometer). Torricelli first solved this problem, but Steiner, a mathematician in the19th century, rediscovered this problem and systematically popularized it, so this point is also called Torricelli point or Steiner point, and related problems are also called Fermat-Torricelli-Steiner problem. The solution of this problem has greatly promoted the development of joint mathematics and has a milestone significance in the history of modern mathematics.