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文[1]中有这样一个结论(题62):设内接于圆O的任一个ΔABC的三条中线AA′、BB′、CC′交于G,它们的延长线和圆O分别交于L、M、N.则AG GL+BG GM+CG GN=定值(等于3).笔者借助几何画板尝试将此结论推广到椭圆,发现依然成立.但在推广到双曲线时,发现当中线所在直线与双曲线相交时,另一交点可能在中线AA′、BB′、CC′的反向延长线上,此时AG GL+BG GM+CG GN并不等于3,但它们的“代数和”(即若
In [1], there is such a conclusion (item 62) that the three center lines AA ’, BB’ and CC ’of any ΔABC inscribed in the circle O are intersected with G and their extension lines and circles O are respectively intersected with L , M, N. AGGL + BG GM + CG GN = set value (equal to 3) .I tried to extend this conclusion to the ellipse by using the geometric drawing board, and found that it still holds.However, when it is extended to a hyperbola, When a straight line intersects with a hyperbola, the other intersection may be on the reverse extension of the midline AA ’, BB’, CC ’, where AG GL + BG GM + CG GN is not equal to 3, but their “ ”(That is, if