本章研究圆组。首先,由一些熟知的初等几何定理,引入“幂”的关系。对这关系的广泛研究引到两个圆的根轴,以及共轴组的性质,然后讨论非常重要的反演理论,并建立它的基本原理。本章包含很多研究三角形时需要的理论.在第5章,对本章涉及内容的某些进一步发展,将更详细地讨论。
首先,我们重新建立 初等几何学中的两个标准的定理,并将它们融合为一个定理,以适合我们的目的。
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定理 如果从一个定点作直线与一个定 圆相交,那么从这定点到两个交点的距离的积是定值设P为已知点,在圆内或圆外均可.令直线PAB与PCD分别交圆于A,B与C,D在每一种情况,△PAD与△PCB的对应角均由相同的弧度量,所以两个三角形相似,
·= ·
证毕。
系如果定圆的半径为r,圆心为O,那么定积·等于
p=-r².
定义 点P关于圆心为O、半径为r的圆的幂是-r²
41 读者可以建立冪的下列性质,它直接依赖于上面的定理与定义。
定理如果点P在圆外,那么 P关于这个圆的幕是正的,并且等于P到这圆的切线的平方。如果P在圆上,这幕是零。.如果P在圆内,这幕是负的;它可以解释为过P的直径被分成的两条线段的积:(+ r)(-r),或过P而且垂直于OP的弦的一半的平方的相反数。
过圆内-点P,垂直于OP的弦,被P点平分,称为P点的最小弦;因为过P的弦中,这条弦最短,如果点在圆外,幂的关系用圆的切线简洁地表示;如果点在圆内,则用最小弦表示。