What is the principle of melon beans and what is the formula of the principle of melon beans?

Gua Bean Principle: If the distance ratio from the two moving points to a fixed point is a fixed value, and the angle is a fixed angle, the motion path of the two moving points is the same. The melon-bean principle is a master-slave linkage trajectory problem.

The active point is called the melon, the driven point is called the bean, the melon moves in a straight line, and the trajectory of the bean is also a straight line. The melon moves in a circle, and the trajectory of the bean is also a circle. The key is to make the motion trajectory of the driven point, and make the special point of the driven point according to the special position point of the active point, so as to form a trajectory.

The double movement point hides the circle.

Model feature: "Fixed-ratio and fixed-angle".

Let's first explain the fixed ratio and fixed angle.

P and Q are moving points, and A is a fixed point. However, P and Q always keep the ratio of line segment PA to QA fixed during the movement, and the angle between line segment PA and AQ is a fixed value.

There are two contents of the melon bean principle:

1. When one of the endpoints of a line segment moves on a certain graph, the trajectory of the midpoint of the line segment is similar to that of this graph. The likeness ratio is 1:2. Of course, other ratios are also available.

When point A moves on the circle O, the midpoint trajectory of AB is also a circle, and the ratio of the radius is equal to 1:2

The melon bean principle is a trajectory problem. The ratio of the distance from two moving points to a vertex is certain, and the angle between these two distances is a point, when the trajectory of one moving point (called the active point, melon) and the other point (called the driven point, bean) is the same as the trajectory shape of the active point. This principle has an application in mechanical transmission.

The melon-bean principle is a dynamic problem - master-slave linkage. When answering, we need to have the idea of trajectory, that is, we must first clarify the trajectory of the active point.

Then it is necessary to figure out the relationship between the active point and the driven point, and then determine the trajectory of the driven point to solve the problem, but when solving the problem, it is necessary to conform to the principle that the solution is not overclassed, so in the end, the knowledge of rotation similarity is still used to solve the problem, that is, the dynamic hand-in-hand model.

Knowledge and methods involved:

Knowledge: similar, triangular.

The sum of the two sides is greater than the third side, the distance from the point to the line is the shortest perpendicular segment, and the point to the point on the circle is collinear has the maximum value.

Method: Step 1: Find the trajectory of the active point.

Step 2: Find the relationship between the driven point and the active point.

Step 3: Find the start and end of the active point.

Step 4: Determine the trajectory of the driven ruler cave point by similarity.

Step 5: Determine the maximum value of the dot line and dot circle according to the trajectory.

The principle of melon and beans is "plant melons and get melons, plant beans and get beans", also called "Pengcheng principle".

Specifically, if the distance ratio from the two moving points to a fixed point is a fixed value, and the included angle is a fixed angle, the motion path of the two moving points is the same. The melon moves in a circle, and the trajectory of the bean is also a circle.

The key is to make the movement trajectory of the driven point, and make the special point of the driven point according to the special position point of the active chaos point, so as to form a trajectory.

Conclusion: 1. The trajectory of C is the same as that of B, which is a circle.

2. The angle between the corresponding line segments on the B circle and the C circle is equal to A.

3. ab ac is a fixed value k.

4. The ratio of the length of the C motion to the length of the B motion is equal to K.

5. The ratio of the radius of circle B to the radius of circle C is k.

6. If AB is not equal to AC, then there is ABM AM'c, similarity ratio. for k.

The melon bean principle was proposed by Zhang Jianquan.

The melon bean principle is also called the Pengcheng principle: if the distance ratio from the two moving points to a fixed point is a fixed value, and the angle is a fixed angle, then the motion path of the two moving points is the same. The melon-bean principle is a master-slave linkage trajectory problem.

The active point is called melon, and the driven point is bean. The melon moves in a straight line, and the trajectory of the bean is also a straight line. The melon moves on a circle, and the trajectory of the bean is also a circle.

The key is to make the movement trajectory of the driven point, and according to the special position of the active point, make the special point of the driven point so as to form a trajectory.

Mathematics: Mathematics is the study of concepts such as quantity, structure, change, space, and information.

Mathematics is a radical method of human referential use to strictly describe the abstract structure and pattern of things, and can be applied to any problem that is lacking in the real world, and all mathematical objects are inherently artificially defined. In this sense, mathematics belongs to the formal sciences.

And not natural science. Different mathematicians and philosophers have a range of opinions on the exact scope and definition of mathematics.

in the history of mankind.

Mathematics plays an irreplaceable role in development and social life, and it is also an indispensable basic tool for learning and researching modern science and technology.

The above content refers to Encyclopedia - Mathematics Annihilation.

The melon bean principle is a dynamic problem - talking about the master-slave linkage.

When solving the problem, it is necessary to have the trajectory idea, that is, to first clarify the trajectory of the active point, and then to figure out the relationship between the active point and the driven point, and then determine the trajectory of the driven point to solve the problem, but when solving the problem, it is necessary to conform to the principle that the solution is not overclass, so in the end, the problem is solved by using the knowledge of rotation similarity, that is, the dynamic hand-in-hand model.

Knowledge and methods involved:

Knowledge: Similarity, the sum of the two sides of a triangle is greater than the third side, the distance between the point and the line is the shortest, and the point to the point on the circle is collinear has the maximum value.

Method: Step 1: Find the trajectory of the active point.

Step 2: Find the relationship between the driven point and the active point.

Step 3: Find the start and end points of the active point.

Step 4: Determine the trajectory of the driven point by similarity.

Step 5: Determine the maximum value of the dot line and dot circle according to the trajectory.

Consider: When point p is moving on a circle 0, what is the trajectory of point q?

Analysis] Observation** shows that the trajectory of the point q is a circle, and what we need to determine is what is the relationship between this circle and the circle 0?

Considering that the Q point is always the AP midpoint, connect A0 and take the A0 midpoint M, then the M point is the center of the Q point trajectory circle, and the radius Mq is half of the OP, and at any time, there is AMQ-AOP, Qm:PO=AQ:AP=1:2.

Summary] Determining the trajectory circle of the Q point is to determine its center and radius, which can be obtained from AQP always collinear: AM0 three-point collinear, and Q is the midpoint of AP: AM=1 The point trajectory is proportionally scaled to the P point trajectory.

According to the relative position relationship between the moving points, the relative position relationship of the circle center is analyzed. According to the quantitative relationship between the moving points, the quantitative relationship of the radius of the trajectory circle is analyzed.

Mathematical application of the melon bean principle:

The melon-bean principle can be mathematically rigorously proved, and if it is not strictly proven, it can also be explained by holistic thinking and equivalence thinking. The first is the dialectical relationship between the individual and the whole, which is composed of multiple individuals, for example, a straight line or a circle is composed of multiple points and is empty. In the melon-bean problem, a single moving point is an individual, and the trajectory (straight line segments, circles, polygons) is a guessing whole.

The mantra of the principle of melon beans is "resistant to planting melons and getting melons, and planting beans in sedan chairs to get beans". The principle of melon beans comes from the idiom "plant melons and get melons, plant beans and get beans", which is a metaphor for what you do and what you get to get the results of making fun of acres.