If the connecting lines of the corresponding vertices of two triangles intersect at a point, the intersection points of the corresponding edges of the two triangles must be on the same straight line.

(If the intersections of the corresponding sides of two triangles are on the same straight line, the connecting lines of the corresponding vertices of the two triangles must intersect at one point. )

The six vertices of a hexagon are on a quadratic curve if and only if the intersections of three pairs of opposite sides are on the same straight line.

If and only if the straight lines connecting the three pairs of vertices intersect at one point, the six sides of the hexagon cut the quadratic curve. )

2. Duality rules in physics

In electromagnetism, there is a dual relationship between the electrostatic field in a uniform medium and the constant electric field in a uniform conductive medium. The electric displacement vector D and the current density vector J are dual, and the charge Q and the current I are dual ... There are dual relationships between voltage source and current source, short circuit and open circuit, series and parallel connection, resistance and conductance, capacitance and inductance in the circuit. When the node voltage method and the loop current method are used, the dual equations with the same form will be obtained without changing the values of dual elements, and the same set of solutions will be obtained.

3. Duality rules in modern control theory.

In automatic control theory, it is sometimes necessary to study the controllability and observability of the system. Using duality principle can bring a lot of convenience to the study of system equations.

Application:

Duality has certain application in finding the sum or product of several terms in a sequence. If we can analyze the symmetry of its structure and combine the symmetrical beauty of mathematics with the conditions or conclusions of the topic, we can construct a set of interrelated dual formulas, thus determining the overall thinking or starting direction of solving the problem. Its essence is to make the revelation of beauty and the pursuit of beauty become the macro-guiding force in the process of solving problems, making the process of solving problems more concise and vivid.

Symmetry is often manifested in mathematics as the symmetry of numbers or figures, the duality or correspondence of propositions or structures. In the process of solving mathematical problems, if we can actively explore the implicit symmetry in the problem and skillfully use symmetry, we can make complex problems clear and understand, and we can turn the difficult into the easy and simplify the complicated.