As shown in the figure, in △ABC, p is any point. Connect AP and BP to get △ABP.
Combined atlas
Combined Atlas (2 sheets)
With point B as the rotation center, rotate △ABP counterclockwise by 60 to get △EBD.
∵ rotate 60, BD=BP,
△ DBP is an equilateral triangle.
∴PB=PD
So PA+PB+PC=DE+PD+PC.
It can be seen that when E, D, P and C are * * * lines, PA+PB+PC is the smallest.
If lines e, d and P***,
∵ equilateral delta DBP
∴∠EDB= 120
Similarly, if the line of d, p and c is * * *, then ∠ CPB = 120.
Point ∴P is the point that satisfies ∠ APB = ∠ BPC = ∠ APC = 120.
Historical background
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.