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2014年全国高中数学联赛陕西赛区预赛的解几题甚是有趣,笔者研究发现,该问题可以拓广到一般情形,本文给出该问题在圆和椭圆中的拓广形式.问题:如图,已知圆O:x2+y2=1与x轴交于A、B两点,与y轴交于点C,M是圆O上任一点(除去圆O与两坐标轴的交点).直线AM与BC交于点P,直线CM与x轴交于点N,设直线PM、PN的斜率分别为m、n,求证:m-2n为定值.
2014 national high school mathematics league in Shaanxi Division preliminaries is very interesting solution, the author found that the problem can be extended to the general situation, this paper gives the problem in the circle and ellipse in the expansion of the problem: the problem is: Figure, Known circle O: x2 + y2 = 1 and the x-axis at the A, B two points, and the y-axis at the point C, M is any point on the circle O (except circle O and the intersection of the two axes) BC to point P, the intersection of line CM and the x-axis at point N, set the slope of the line PM, PN were m, n, verify: m-2n for the set value.