definition

In a polygon, the point with the smallest sum of distances to each vertex is called the fermat point of the polygon.

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.

(1) In an equilateral triangle, BP=PC=PA, and BP, PC and PA are the heights of the three sides of the triangle and the bisector of the triangle respectively. Is the center of inscribed circle and circumscribed circle. △BPC?△CPA?△PBA .

(2) When BC=BA but CA≠AB, BP is the bisector of the height and the median line on the triangle CA and the angle on the triangle.

certificate

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

△CC 1B and △AA 1B, BC = ba 1, Ba = bc 1, ∠ CBC 1 = ∠ b+60 degrees = ∠ ABA/kloc.

△CC 1B and △AA 1B are congruent triangles, and ∠PCB=∠PA 1B is obtained.

In the same way, ∠CBP=∠CA 1P can be obtained.

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.

And ∠BPA= 120 degrees, so a, p and d are on the same straight line.

And ∠CPB=∠A 1DB= 120 degrees, ∠PDB=60 degrees, ∠PDA 1= 180 degrees, so a, p, d, a.

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

Take any point M (not coincident with point P) in △ABC, connect AM, BM and CM, rotate △BMC 60 degrees around point B to coincide with △BGA 1, connect AM, GM, A 1G (same as above), and then AA 1

Plane quadrilateral fermat point

Fermat point's proof in a quadrilateral is easier to learn than fermat point's proof in a triangle.

(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).

Fermat once asked an interesting question about triangles: find a point on the plane where the triangle is located to minimize the sum of the distances from that point to the three vertices of the triangle. That is, finding a point P in ABC minimizes the value of PA+PB+PC, which is called "fermat point".

Today we are going to explore fermat point. First of all, triangles are divided into two situations:

(1) When the internal angle of a triangle is greater than or equal to 120 degrees, the fermat point is the vertex of the internal angle.

Let's verify this conclusion: for any point P in the triangle, extend BA to C' to make AC=AC', make ∠C'AP'=∠CAP, make AP'=AP, PC'=PC, that is, rotate the triangle APC around A.

Then △ APC △ AP' c' (rotation invariance)

∵∠ BAC ≥ 120 (known)

∴∠ PAP' = 180-∠ BAP-∠ C 'AP' (meaning of a straight angle) = 180-∠ BAP-∠ CAP (equivalent substitution) = 65436.

∴pap' (called AP'=AP) in the isosceles triangle, AP ≥ PP' (∠ PAP' < ∠ APP').

∴pa+pb+pc≥pp'+pb+ p ' c ' & gt； BC' (the sum of two sides is greater than the third side) =AB+AC (known AC=AC')

So A is fermat point. This is the previous conclusion.

Let's discuss the second situation:

(2) If all three internal angles are within 120 degrees, then fermat point is the point that makes the connecting line between fermat point and the three vertices of the triangle form an included angle of 120 degrees.

Make a point P in △ABC, so that ∠ APC = ∠ BPC = ∠ CPA = 120, which is the perpendicular line of PA, Pb and PC, and intersect with three points D, E and F (as shown in the figure), and then make a point P', which is not coincident with the point P, and connect P'.

∫≈APB = 120 ,∴∠pab+∠pba= 180- 120 = 60

And ∠ PAF = ∠ PBF = 90 and ∴ F = 180-(90+90-60).

In the same way, we can get: ∠ d = ∠ e = ∠ f = 60, that is, △DEF is an equilateral triangle, let the side length be d and the area be s.

Then S= 1/2 d (PA+PB+PC)

∫P ' h≤P ' a

∴ 1/2×d×p ' h×2s≤ 1/2×d×p ' a×2s

∫ 1/2×d×p ' h =△EP ' f∴2s△EP ' f≤d×p ' a×s

Similarly: 2S△DP'F≤d ×P'B×S, 2s △ EP'd ≤ d× p 'c× s.

Add up to get 2s (△ EP 'f+△ DP 'f+△ EP'd) ≤ d× s (P 'a+P 'b+P 'c).

∫EP ' f+△DP ' f+△EP ' d =△EDF。

Two sides of 2S×S ≤ d ×S (P'A+P'B+P'C) are divided by s to get 2S ≤ d (P'A+P'B+P'C).

Substitute S= 1/2 ×d (PA+PB+PC) into the above formula to obtain:

PA+PB+PC≤P'A+P'B+P'C, if and only if p and p' coincide, take the equal sign.

So P is fermat point, which is consistent with the above conclusion.

After the above deduction, we can find the fermat point in the triangle:

When the internal angle of a triangle is greater than or equal to 120 degrees, fermat point is the vertex of this internal angle; If all three internal angles are within 120 degrees, then fermat point is the point that makes the connecting line between fermat point and the three vertices of the triangle form an included angle of 120 degrees.

Pierre de Fermat (1601-1665) is a French mathematician and physicist. Fermat has never received a special mathematics education in his life, and mathematics research is just a hobby. However, in France in the17th century, no mathematician can match it. He is one of the inventors of analytic geometry; The main founder of probability theory; And/kloc-people who inherited the world of number theory in the 0/7th century. Fermat, the master of mathematics, was the greatest mathematician in France in the17th century. In particular, his Fermat's Last Theorem has puzzled the wise men in the world for 358 years.

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