First, study motivation.
In the next 265,438+0 century, Kaohsiung will follow the footsteps of Taipei, the capital, to build a rapid transit system and develop Haidu Kaohsiung into the most advanced metropolitan area. Different from Taipei, Kaohsiung MRT adopts underground buildings, in which the basic road network of red line and orange line is planned. According to my father, no matter which route, you need to build a MRT main engine factory. The main engine factory is equivalent to the heart of the MRT for human beings, so I thought: Can you find a location with the smallest distance to each MRT station for easy control?
It is also known from the literature that the sum of the distances from one point to three vertices in a triangle is the smallest, which is called "fermat point". Therefore, from this perspective, a series of discussions and studies have been made on the nature of fermat point.
Second, the purpose of the study
(1) Prove the existence and characteristics of fermat point by mathematical methods.
(2) Explore fermat point's related theories by using physical methods.
(3) Find the experimental results of fermat point verification physics in rectangular coordinate system.
(4) Explore the application examples of fermat point in life.
Third, research equipment.
Pulley, batten, cotton thread, mud block, square paper, protractor.
Fourth, the research process
(a) prove the existence and characteristics of fermat point by mathematical methods;
I. In fact, some famous mathematicians have put forward relevant methods and proofs before. I listed all the methods found in the literature in the appendix, and I also tried to do so to see if there are other ways to find fermat point:
1. fermat point's solution (refer to figure 1).
(1) makes one or three internal angles of △ABC less than 120.
(2) Take as one side and make regular triangles △ABD and △ACE outward respectively.
(3) Connect and pay at point P, then point P is what you want.
2. The nature of fermat point: L = ++ is the minimum value.
~ First of all, prove the existence of fermat point made by the above practices-
(see figure 2) rotate △BPC,
Make consistent with (=),
Point p falls on point H.
Then ∠ BPC = ∠ BHG = 120.
ピ BHP Billiton = 60 (confirmed in)
∴∠BHG+∠BHP= 180
So, a, p, h, g three-point line.
ㄇ.*△BHG△BPC
Get = =
∠∠2+∠3 = 60 and∠∠1= ∠ 3.
∴∠ 1+∠2=60 =∠PBH
Therefore △BPH is positive△, so =
Knowing a little p makes++=+=
~ then prove that the distance from this point to three vertices is the smallest.
(Refer to Figure 3) Take another point in ABC, where q is different from p,
Connect,
ピ. Reference to the proof method in step (1) can also prove++=++.
ㄇ.
So point P makes++the minimum.
Ⅱ. fermat point's discussion is generally confined to the interior of triangles whose triangles are all less than 120. If we discuss any triangle with an angle greater than or equal to 120, can we find the shortest distance from one point to three vertices? (See Figure 4)
△ ABC's (1) ∠ A > 120, and p is any point in△ ABC.
Extend to b', make =
Do ∠B'AP'=∠BAP，take =
So △B'AP' △BAP, get =
So++=++,
(2) because < a >120, < b' ab < 60,
Also < ∠PAP' < 60；; So the isosceles triangle P'AP
Medium ∠ AP 'p > 60, so >
Then++>++>+,which is++>+
That is to say, if a point P coincides with A, then P is the point with the smallest sum of distances to A, B and C. ..
(3) It is proved that if an internal angle of a triangle is known to be greater than or equal to 120, then fermat point is the vertex of the internal angle.
Ⅲ. Only three triangles with internal angles less than 120 exist fermat point, but in daily life, it is not just triangles that need to require the minimum distance of each vertex! That is to say, if we can find a point P after changing the shape to minimize the sum of the distances from the point P to the vertex, we will discuss the simplest quadrilateral first (refer to Figure 5).
(1) Known: quadrilateral ABCD
Find: point p in ABCD
Exercise: in quadrilateral ABCD
The diagonal is a straight line.
The diagonal is the smallest distance between a and C.
Similarly, the diagonal is the minimum distance between b and d.
It is found that the intersection point p of is a point in the quadrilateral ABCD, so that++is the minimum value.
That is to say, the sum of the distances from point P to the four vertices of the quadrilateral is the smallest.
(2) Proof: (Refer to Figure 6)
Take another point p' in the quadrilateral ABCD, which is different from P.
Connect,,,
△P'BD，△AP'C
+> and+> (the sum of any two sides is greater than the third side)
∴ + + + > + = + + +
So point P makes++the minimum.
(B) the use of physical methods to explore fermat point's theory-often heard that "mathematics is science.".
Mom, can you use scientific methods to verify the existence of fermat point or some properties of fermat point? After consulting the teacher's opinion and thinking, I did a series of mechanical experiments:
1. Experiment 1: Prove the nature of fermat point from the balance of three forces-between three angles, and it must be 120.
(1) Assemble a regular triangle with battens as edges, and install a pulley at each of the three vertices. Take three cotton threads of equal length, one end of which is hung with a mud block W of equal weight, and the other end is connected together to represent point P. ..
(2) Let the weight droop naturally and reach a static state, and measure the angles of ∠APB, ∠APC and ∠BPC (see table 1 for data).
(3) Because the weights of the three weights are equal, the tensions of the three lines are the same, that is, when F 1=F2=F3=W, the "closed triangle (refer to Figure B)" formed by the force diagram (refer to Figure A) is a regular triangle, that is, three angles sandwiched by the three forces.
Both are 120.
(4) Place the experimental device in step (2) vertically above a coordinate plane, record the coordinates of point P, and then sum the linear functions of point P obtained in step (3), and use computer program to calculate (see Annex 2 for details) whether they are consistent.
(5) Repeat the above steps for 5 times, and change the shape of the triangle to repeat the operation.
2. Experiment 2: It is found that fermat point has the lowest potential energy.
(1) Assemble a regular triangle ABC with battens as sides and put it on a horizontal plane. Each of the three vertices is equipped with a pulley. One end of three equal-length cotton threads is hung with an equal-weight mud block W, which is suspended by three pulleys respectively. According to the experiment 1, the point P is fermat point.
(2) Hang a mud block W at point P (fermat point) to make the weight naturally move vertically downwards to reach a static state (see Figure C for the device), measure the vertical distance between point P and the horizontal plane at this time, and take the average of three times, and the height is hP.
(3) Move point P to any point on three sides, three sides and three sides at will, and then release the heavy object, and find that it will tend to fermat point on either side. According to the principle that an object will move freely to the lowest point of energy, it can be proved that fermat point has the lowest potential energy.
(4) Record the experimental process of step (3) and get the potential energy height h' (cubic average), (representing the situation after being released from the point, and so on),,, (see Table 2 for data).
(5) Repeat the above steps for 3 times, and change the shape of the triangle to repeat the operation.
(3) Verify the experimental results of fermat point in Cartesian coordinate system-Cartesian coordinate system is often used in map representation. Can you find the point P in Cartesian coordinate system (point P is the smallest distance to each vertex) and then verify our experimental results with computer programs?
(1) triangle-
A. For convenience, fix it on the X axis with the vertex as the origin, by using the idea mentioned in (1) above.
B: Let's discuss special triangles one by one, and then generalize them to general triangles.
ⅵ. Triangle (see Figure 7)
=
= =
Therefore, the coordinate of point P is ()
ⅶ. isosceles triangle (see figure 8)
The quadrilateral AOBC is kite-shaped.
∴
∠ OPC = 120。
Therefore ∠ OPD = 60.
Therefore = = =
The coordinate of point P is ()
Right triangle (see Figure 9)
Let the function passing through point p be y = ax+b.
Substitute four coordinates a, b, c and d.
Find the equation and solve the simultaneous equation.
:y= x+y 1
:y= (x-x 1)
The coordinates of point p are
Isosceles right triangle
The isosceles right triangle is a kind of isosceles triangle, so the coordinates of point P can refer to the solution of isosceles triangle. Similarly, the coordinates of point P can also refer to the solution of right triangle.
ヵ Arbitrary triangle (refer to Figure 10)
Let the function passing through point p be y = ax+b.
Substitute four coordinates a, b, c and d.
Find and solve simultaneous equations
:y=
:y=
The coordinates of point p are
(2) Quadrilateral-
A. For convenience, fix it on the X axis with the vertex as the origin by using the idea mentioned in (3) above.
B: First, discuss the special quadrilateral one by one, and then extend it to any quadrilateral.
ⅵ. Square (refer to Figure XI)
∵ quadrilateral ABCO is a square.
Equal sum =
(Diagonal lines in a square are equally divided)
Therefore, the coordinate of point P is ()
ⅶ. Rectangular shape (see figure 12)
∵ Quadrilateral ABCO is a rectangle.
Equal sum =
(Diagonal lines in a rectangle are equally divided)
Therefore, the coordinate of point P is ()
ミ. parallelogram (see figure 13)
∵ Quadrilateral ABCO is a parallelogram.
Equal sum =
(Diagonal lines in parallelogram are equally divided)
Therefore, the coordinate of point P is ()
リ. Diamond (see Figure 14)
The quadrilateral ABCO is a diamond.
Equal sum =
(Diagonal lines in the diamond are equally divided)
Therefore, the coordinate of point P is ()
It is found that if the diagonal of a quadrilateral is equally divided,
Then its point P is the midpoint of the diagonal of this quadrilateral.
ヵ. isosceles trapezoid (see figure 15)
////
∵ // //
∴△ABP~△OPC
Set to, such as y-
( )
∴
ABCO is an isosceles trapezoid.
∴
Therefore, the coordinate of point P is ()
12. A trapezoid with two right angles (see figure 16).
////
∵ // //
∴△ABP~△OPC
Set to, such as y-
( ) =
∴
∵ // //
∴△ADP~△AOC
Start doing it with great energy; Start resolutely
∴
Therefore, the coordinate of point P is ()
Arbitrary trapezoid (see figure 17)
////
∵ // //
∴△ABP~△OPC
Set to, such as y-
( )=
∴
∵ // //
∴△ADP~△AOC
Start doing it with great energy; Start resolutely
∴
Therefore, the coordinate of point P is ()
(For convenience, the two vertices are fixed on the X axis. )
ⅵ. kite shape (see figure 18)
It's diagonal
Point p is on the x axis.
The quadrilateral ABCO is kite-shaped.
equal division
Therefore, the coordinate of point P is ()
Arbitrary convex quadrilateral (see figure 19)
Let the function passing through point p be y = ax+b.
Substitute four coordinates a, b, c and o.
Find the equation and solve the simultaneous equation.
:y=
:y=0
The coordinates of point p are
Research achievements of verbs (abbreviation of verb)
Using rectangular coordinates and physical experiments, according to the different shapes of graphics, the data needed by physics and mathematics are analyzed one by one to find out their correlation.
(1) P-point linear function of regular triangle, right triangle, isosceles triangle, isosceles right triangle and any acute triangle (see the aforementioned rectangular coordinate diagram):
1. Regular triangle: ()
2. isosceles triangle: ()
3. Right triangle:
4. isosceles right triangle: () or
5. Arbitrary triangle:
(2) P-point linear function of square, rectangle, parallelogram, rhombus, isosceles trapezoid, trapezoid with two internal angles at right angles, arbitrary trapezoid, kite shape and arbitrary quadrilateral (refer to the diagram of rectangular coordinates mentioned above);
1. Square, rectangle, parallelogram and diamond: ()
2. isosceles trapezoid: ()
3. A trapezoid with two right angles inside: ()
4. Arbitrary trapezoid: ()
5. Kite shape: ()
6. Arbitrary quadrilateral:
(3) Data of experiment 1 (table 1):
Regular triangle A (2,2) B (4,0) C (0,0)
Multiply by 1 2 3 4 5
Angle ∠ APC120120.5118120.5120.
∠APB 120 1 19.5 122 12 1 124
∠BPC 120 1 19 19 120 1 17
The coordinates of point P are (1.96, 0.9 1) (2.00, 1.34) (2.06, 1.40) (2.05,1.44) (.
The calculated value is about (2, 1. 15470 1).
Isosceles triangle A( 1.5,) b (3,0) c (0,0)
Multiply by 1 2 3 4 5
Angle ∠ APC12311819122120.
∠APB 1 19 120 1 19.5 120 120
∠BPC 12 1 120 12 1 12 1 1 19
The coordinate measurement of point P is (1.50,1.08) (1.51.92) (1.61.89) (/kloc-0)
The calculated value is about (1.5, 0.866025).
Right triangle a (0,4) b (3,0) c (0,0)
Multiply by 1 2 3 4 5
Angle ∠ APC11911221.5121.
∠APB 120 12 1 120.5 12 1 120
∠BPC 120 12 1 19.5 1 18 120
The coordinates of point P are (0.66,0.84) (0.69,0.65) (0.79,0.55) (0.71,0.58) (0.84,0.56).
The calculated value is about (0.75117,0.789).
Isosceles right triangle A (0,3) B (3,0) C (0,0)
Multiply by 1 2 3 4 5
Angle ∠ APC1211201918.5120.
∠APB 1 19 1 19.5 12 1 1 19 1 19.5
∠BPC 1 18 18 1 19.5 1 19.5 19.5 1 18
The coordinate metric of point P is (0.58, 0.6 1) (0.72, 0.58) (0.62, 0.59) (0.50, 0.80) (0.66, 0.56).
The calculated values are about (0.633975, 0.633975).
Any triangle A (2.2,3.6) B (4.8,0) C (0,0)
Multiply by 1 2 3 4 5
Angle ∠ APC121.519.5120120.
∠APB 1 19.5 120 120.5 120 122
∠BPC 1 19 120.5 120 1 19 120
The coordinates of point P are (1.96, 0.9 1) (2.00, 1.34) (2.06, 1.40) (2.05,1.44) (.
The calculated value is about (2.257 189, 1.5438+0958).
Results: The measured value is very close to the calculated value, which proves that the point P obtained in the experiment is fermat point, but it will be affected by friction and other factors in the experiment, resulting in errors.
(4) Data of Experiment 2 (Table 2):
Regular triangle A (2,2) B (4,0) C (0,0)
Multiply by 1 2 3
Point p coordinates P (2. 12,1.31) (1.92,1.31) (1.96,/kloc.
(2.08, 1.33) (2. 15, 1. 10) ( 1.95, 1.09)
( 1.98,2.06) ( 1.94, 1. 13) ( 1.86, 1. 13)
(2.3 1, 1. 13) (2. 15, 1. 17) ( 1.98,0.97)
The original vertical height (hP) is 35.3cm.
The height (h') of fermat point is 32.25 cm.
32.35cm 31.6cm32.1cm
32.25cm 31.8cm 32.5cm.
32.4cm31.75cm 32.0cm.
Isosceles triangle A( 1.5,) b (3,0) c (0,0)
Multiply by 1 2 3
Point p coordinates p (1.54,0.90) (1.56,0.97) (1.72,0.91).
( 1.44,0.69) ( 1.52,0.80) ( 1.5 1,0.92)
( 1.50,0.66) ( 1.6 1,0.74) ( 1.64,0.68)
( 1.46,0.77) ( 1.48,0.65) ( 1.43,0.74)
The original vertical height (hP) is 35cm.
The height (h') of fermat point is 32.3 cm.
3 1.9 cm 32. 1 cm 32.0 cm
32. 1 cm 32.2cm 32.15cm
32.2 cm 32.3 cm 32.25 cm
Right triangle a (0,4) b (3,0) c (0,0)
Multiply by 1 2 3
Point p coordinates P (0.84, 0.74) (0.86, 0.64) (0.74, 0.69)
(0.96,0.68) ( 1.00,0.62) ( 1.06,0.67)
(0.84,0.68) (0.79,0.85) (0.84,0.80)
(0.78,0.54) (0.73,0.74) (0.86,0.79)
The original vertical height (hP) is 35.2cm.
The height (h') of fermat point is 32.4 cm.
32.2 cm 32.25 cm 32.3 cm
32.4 cm 32.6 cm 32.55 cm
32.15cm 32.5cm 32.4cm
Isosceles right triangle A (0,3) B (3,0) C (0,0)
Multiply by 1 2 3
Point p coordinates P (0.64, 0.6 1) (0.66, 0.53) (0.62, 0.66)
( 1. 12,0.59) ( 1.06,0.54) (0.95,0.58)
(0.75,0.74) (0.87,0.68) (0.84,0.88)
(0.76,0.66) (0.92,0.57) (0.78,0.82)
The original vertical height (hP) is 35.3cm.
The height (h') of fermat point is 32.5cm.
32.4 cm 32.2 cm 32.5 cm
32.4 cm 32.6 cm 32.45 cm
32.65 cm 32.5 cm 32.6 cm
Any triangle A (2.2,3.6) B (4.8,0) C (0,0)
Multiply by 1 2 3
Point p coordinates P (2.20, 1.49) (2.42,1.37) (2.1.65).
(2.55, 1.57) (2.65, 1.40) (2.46, 1.27)
(2.26, 1.43) (2.47, 1.4 1) (2.44, 1.29)
(2.55, 1.40) (2.49, 1.48) (2.46, 1.23)
The original vertical height (hP) is 35.5cm.
The fermat point height (h') is 3 1.5 cm.
31.5cm 31.35cm 31.65cm.
31.3cm31.6cm31.5cm.
31.55cm 31.4cm31.5cm.
Results: There is an error between the P point obtained in this experiment and the calculated value of fermat point, which is due to the influence of friction and other factors in the experiment.
Discussion and application of intransitive verb
(1) In a triangle with an angle greater than or equal to 120, it is impossible to find the position of fermat point as a regular triangle by graphic method, because the point P used for graphic method will fall outside the triangle, and the three angles formed by the straight line that does not conform to the point P to the three vertices are all equal to 120. According to the proof, the sum of fermat point is greater than or equal to/kloc-. So there is a triangle whose angle is greater than 120 or equal to it, which is not discussed here.
(2) Although the diagonal intersection of the concave quadrilateral is on the outside, its diagonal and P points are the same as those of the convex quadrilateral, and its practices and coordinates are the same, so it is omitted and not repeated.
(3) From Experiment 2, we found a physical property of fermat point: "fermat point is the point with the lowest energy in the triangle". Because in the second experiment, no matter where the point P moves, it will always move to the origin P after release, which can prove that the origin P is the position with the minimum energy of the triangle.
(4) The measured value obtained in the experiment is very close to the calculated value, which can prove that the point P obtained in the experiment is fermat point, but it will be affected by friction and other factors in the experiment, resulting in errors.
(5) fermat point is also widely used in daily life. As long as it exists between three points and the sum of distances is minimum, it can be applied to the properties of fermat point. For example, how to set up substations in three cities, how to set up high-voltage towers to reduce the waste of electricity, or how to dig a well among three households are all examples of fermat point's application.
(6) It is expected that other physical and chemical methods will be used to prove the nature of early fermat point. These methods are as follows:
1. Try to prove fermat point by electrical method;
In electricity, the resistance is proportional to the length of the wire. If the distance from Fermat point to three vertices of triangle is used to measure the resistance wires with three lengths respectively, is the resistance after parallel connection smaller than that after non-fermat point parallel connection? This paper tries to find out the relationship between three parallel resistors and fermat point.
(1) Take a wire and a resistance wire in series, and then connect three devices in parallel to the same power supply device.
(2) Take the intersection of the conductor and the resistance wire of the device as the vertex of the triangle, and assemble it into a regular triangle with wooden strips.
(3) Connect the ends of three resistance wires to point P in the triangle, then connect them in series with an ammeter and return to the power supply device to form a path.
(4) Turn on the power supply, move the position of point P, find out the place where the current value of point P is the maximum, then put the experimental device vertically above a coordinate plane, record the coordinates of point P, then sum the linear functions of point P obtained in (3), and use a computer program to calculate (see Annex II for details) whether they are consistent.
(5) Change the shape of the triangle and repeat the operation.
2. Try to discuss the essence of fermat point from the viewpoint of theoretical chemistry;
In the development of theoretical chemistry, computer programs can be used to simulate the structural state of small chemical molecules and obtain the most stable (minimum energy) molecular arrangement. For some triangular molecules (such as cyclic ethylene, cyclic ether, cyclic ethylamine), the arrangement of these molecules and external atoms is bound to maintain a stable state with minimum energy in the natural state, and whether the relative position between these atoms is related to the fermat point in the triangular molecular configuration, I want to use something simpler.
Seven. conclusion
There are quite a few proofs about fermat point. This time, in addition to finding some related results in mathematics, we also discussed the physical meaning of fermat point with experiments and rectangular coordinates, which fully proved that "mathematics" is really the "mother of science" again! !
(1) fermat point has two properties in mathematics: "all three included angles are 120" and "the sum of the three vertices is minimum". In addition, in physics, he has the property that fermat point is the lowest energy point in a triangle.
(2) For an isosceles triangle whose vertex angle is less than 120, the fermat point must be at the height of the base, and when the length of the base is the same, fermat point is the same point. And the "fermat point" of the quadrilateral is the intersection of diagonal lines.
(3) According to the theory, the three angles contained in the line segment from Fermat to the three vertices are all 120, which is exactly the same as the three-force angle in the three-force balance, so we can find the position of fermat point through the three-force balance experiment in physics.
Eight. References and others
booklist
1. Author Qiaoxiong Sakai, (1992) Interesting experiment in mechanics, published by Yadong Bookstore.
2. Zhang Jingzhong (1990), The Mathematician's Vision, published in nine chapters.
3. Zhang Dianzhou and Dai Zaiping (1996), Middle School Mathematics in Life, published in nine chapters.
4. Huang Jiali (1997), The Pearl of Geometry, published in nine chapters.
not reveal one's name