Overview:

Fermat's last theorem is introduced, and the properties of fermat point in triangle and fermat point in four parallel sides are deduced.

Introduce Fermat (160 1 ~ 1665), Pierre de French. 160 1 Born in a leather merchant's family near Toulouse. He studied law, worked as a lawyer and was a member of Toulouse local council. He loves mathematics and spends almost all his spare time studying it. Because of his talent and tenacious research spirit, he has achieved fruitful results in many fields of geometric optics and mathematics, and is known as the "king of amateur mathematicians". Most of Fermat's work spread around the world through his letters to friends. He published only a few papers in his life, most of which were published after his death.

In mathematics. Fermat put forward many important theorems, the most famous of which are Fermat's Last Theorem and Fermat's Little Theorem.

Many propositions put forward by Fermat in number theory were not proved by him, but most of them were proved positive by mathematicians in18th century.

In Fermat's time, calculus had not been established, but Fermat had the bud of infinitesimal analysis thought. His method of finding the extreme value of the function, the tangent of the curve and the area of the curved trapezoid formed by the curve and the coordinate axis is quite close to the method described in the current mathematical analysis. It can be seen that Fermat is also a pioneer of calculus.

For natural phenomena, Fermat proposed principle of least action. This principle holds that the occurrence of various phenomena in nature consumes very little energy. Fermat explains the form of honeycomb eye with the principle of minimum action, which is more reasonable than any other form in saving wax consumption. He also excellently applied this principle to the refraction of light. Fermat thinks that when the ratio of light speed in two different media is equal to the ratio of sine of incident angle and sine of refraction angle, the total resistance of the media is the smallest. This provides an important basic principle for geometric optics.

Fermat is a pioneer of combinatorial theory. The concept of "mathematical expectation" put forward by him and Pascal laid the foundation for the development of probability theory and opened up a broad field for the application of this subject.

Fermat point definition

In a plane triangle:

(1). Three triangles with internal angles less than 120, with AB, BC and CA as sides, make regular triangles ABC 1, ACB 1, BCA 1 on the outside of the triangle, and then connect AA 1.

(2) If the internal angle of the triangle is greater than or equal to 120 degrees, then the vertex of this obtuse angle is the demand.

(3) When △ABC is an equilateral triangle, the outer center coincides with fermat point.

Fermat point in a plane quadrilateral:

(1) In the convex quadrilateral ABCD, fermat point is the intersection point p of two diagonal lines AC and BD.

(2) In the concave quadrilateral ABCD, fermat point is the concave vertex D(P).

The following only deduces the triangles we have learned:

(1) The opposite opening angle of fermat point is 120 degrees.

：

∵ It is known that △ ABC 1, △ AB 1C and △ BCA 1 are positive △

∴C 1b=ab cb=a 1b

∫≈c 1bc =≈c 1ba+≈ABC

∠aba 1=∠cba 1+∠abc

∠abc Gong * * * ∠ c1ba = ∠ CBA1= 60 ゜

∴c 1b=ab CB = a 1b∠c 1ba =∠CBA 1

△c 1bc?△ABA 1(SAS)≈ba 1a =≈bcc 1。

Similarly, △ ACA1△ BCB1(asa) ≈ aa1b = ≈ b1BC.

From ∠PA 1B+∠CA 1P=60 degrees, ∠PCB+∠CBP=60 degrees, so ∠CPB= 120 degrees.

Similarly, ∠APB= 120 degrees, ∠APC= 120 degrees.

(2)PA+PB+PC=AA 1

Rotate △BPC 60 degrees around point B to coincide with △BDA 1 and connect PD, then △PDB is an equilateral triangle, so ∠BPD=60 degrees.

∫∠BPA = 120 degrees, so A, P and D are on the same straight line.

Because ∠CPB=∠A 1DB= 120 degrees, ∠PDB=60 degrees, ∠PDA 1= 180 degrees, A, P, D, A.

(3)PA+PB+PC is the shortest.

It is known that a 1a=ap+bp+pc.

Take any point m in the triangle abc and connect AM, BM and MC.

Turn △ MBC 60 counterclockwise at point B.

Link AM, GM, A 1G

∫△PBC?△CGA 1

∴bp=ga 1

∠∠BCA 1 = 60，∠pcb=∠gca 1

∴∠mcg=60

So mc=mg

∴aa 1<； A 1G+GM+MA=AM+BM+CM

Therefore, the distance from Fermat point to the three vertices A, B and C is the shortest.

References: Baidu Encyclopedia, Mathematics Book.