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本文给出两个与圆有关的数学问题的别证,并将它们拓广到圆锥曲线中去,仅供参考.数学问题2095(见文[1])如图1,设⊙O的两条互相垂直的直径AB、CD,E在BD上,AE与CD交于点K,CE与AB交于点L,求证:(EK/AK)2+(EL/CL)2=1.问题提供人给出的解答中运用了正弦定理,同时运算较为繁琐.其实运用平行线分线段成比例定理及勾股定理即可证得结论.
In this paper, we give two different proofs about the circle-related mathematical problems and extend them to the conic for reference purposes only.Mathematical Problem 2095 (see [1]) is shown in Figure 1, where two ⊙O The perpendicular diameters AB, CD, E are on the BD, the intersection of AE and CD is at the point K, and the intersection of CE and AB is at the point L. It is verified that: (EK / AK) 2+ (EL / CL) 2 = 1. The solution given by the use of the sine theorem, while the operation is more cumbersome.In fact, the use of parallel lines sub-line proportional theorem and Pythagorean theorem can be verified conclusions.