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文[1]给出了如下定义:在抛物线中,点D在抛物线的对称轴上且与焦点同侧,直线l与对称轴垂直且与焦点异侧,若点D与直线l到抛物线的顶点等距离,则称点D与直线l为“对偶元素”;在椭圆(双曲线)中,点D在长轴(实轴)所在的对称轴上,直线l与该对称轴垂直且与曲线无交点,若点D与直线l在椭圆(双曲线)中心的同侧,且它们到椭圆(双曲线)中心的距离的乘积为长半轴(实半轴)长的平方,则称点D与直线l为“对偶元素”.文[2]在文[1]基础上给出了圆锥曲线中与顶点有关的对偶元素性质,即如下定理1和定理2(顺便指出,文[2]定理3即定理2):
[1] gives the following definition: In a parabola, the point D is on the same axis of the parabola and is the same as the focal point. The line l is perpendicular to the axis of symmetry and is out of focus. If the point D and the line l reach the vertex of the parabola In the ellipse (hyperbola), the point D is on the axis of symmetry where the long axis (real axis) is located, and the straight line l is perpendicular to the axis of symmetry and is equal to If there is no intersection point between the point D and the line l on the same side of the center of the ellipse (hyperbola) and the distance between them and the center of the ellipse (hyperbola) is the square of the length of the long axis (real semi-axis) On the basis of the paper [1], the properties of the dual elements related to the vertices in the conic are given as follows: Theorem 1 and Theorem 2 (By the way, the paper [ 2] Theorem 3 Theorem 2):