Three kinds of required questions on the principle of melon and bean

The three necessary questions of Guadou principle are: Guadou principle: If the distance ratio between two moving points to a fixed point is a fixed value and the included angle is a fixed angle, the trajectories of the two moving points are the same. Guadou principle is a master-slave linkage trajectory problem. The driving point is called melon and the driven point is called bean. Melons move in a straight line, and so do beans.

The application of the melon-bean principle in mathematics;

The principle of melon and bean can be strictly proved mathematically. If there is no strict proof, it can also be explained by holistic thinking and equivalent thinking. The first is the dialectical relationship between the individual and the whole. The whole is composed of multiple individuals, such as a straight line or a circle composed of multiple points. In the melon-bean problem, a single moving point is an individual, and the trajectory (straight line/line segment, circle, polygon) is a whole.

To understand the principle of melon and bean, we should use holistic thinking, from the linkage correspondence between driving point and driven point to the correspondence and linkage of their motion trajectories. Correspondence: the trajectory of the driving point corresponds to the trajectory of the driven point, such as straight line to straight line and circle to circle; Linkage: the trajectory of the active point is translated &; Spin and potential conversion, the change produces the trajectory of the driving point.

The conclusion of the Guadou principle model is that the trajectory of the driving point and the driven point are similar, and the trajectory of the driven point can be determined according to the included angle formed by the connecting line between the driving point and the driven point and the ratio of the distance between the driving point and the driven point to the fixed point.

If point P is a fixed point and point A is a moving point on circle O, then connecting PA is a line segment PB perpendicular to PA, so that PB is equal to PA. If the locus of point A is a circle, then the locus of point B is also a circle. If point P is a fixed point and point A is a moving point on the straight line L, connect PA as a line segment PB perpendicular to PA, so that PB is equal to PA.