Desargues逆命题证明的射影几何学方法

藉助射影几何的理论,通过将直线投影到无穷远,将两相交直线投影成两平行直线及任意四边形投影成平行四边形。首先给出Desargues逆命题在平面域内的证明,然后用射影几何方法构造了一个辅助三点形,利用Desargues定理证得了两异面三点形对应边的交点共线,再用如上所述平面域内所得的结论证得了两同面三点形对应顶点的连线共点。最终得到了该逆命题在空间域内的证明。

Proof of Desargues converse proposition with projection geometry

At first, with the theory of projective geometry, by the means of projecting straight line onto infinite distance and introducing the projection of two intersecting lines onto two parallel lines as well as arbitrary quadrangle onto parallelogram, it proves the converse proposition of Desargues within the plane area. Then it makes up an auxiliary three-point shape by using the method of projective geometry. With the help of Desargues theorem, that the intersection points of the corresponding sides of two different surface three-point shapes have the same line has been proved. With the above-proved theory, that the ligature of the corresponding vertes of two coplane three-point shape have the same point is therefore proved. Finally, it gets the result of testifying the converse proposition of Desargues within the space.