1976 USAMO 第 2 题

If $A$ and $B$ are fixed points on a given circle and $XY$ is a variable diameter of the same circle, determine the locus of the point of intersection of lines $AX$ and $BY$. You may assume that $AB$ is not a diameter.

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[asy] size(300); defaultpen(fontsize(8)); real r=10; picture pica, picb; pair A=r*expi(5*pi/6), B=r*expi(pi/6), X=r*expi(pi/3), X1=r*expi(-pi/12), Y=r*expi(4*pi/3), Y1=r*expi(11*pi/12), O=(0,0), P, P1; P = extension(A,X,B,Y);P1 = extension(A,X1,B,Y1); path circ1 = Circle((0,0),r); draw(pica, circ1);draw(pica, B--A--P--Y--X);dot(pica,P^^O); label(pica,"$A$",A,(-1,1));label(pica,"$B$",B,(1,0));label(pica,"$X$",X,(0,1));label(pica,"$Y$",Y,(0,-1));label(pica,"$P$",P,(1,1));label(pica,"$O$",O,(-1,1));label(pica,"(a)",O+(0,-13),(0,0)); draw(picb, circ1);draw(picb, B--A--X1--Y1--B);dot(picb,P1^^O); label(picb,"$A$",A,(-1,1));label(picb,"$B$",B,(1,1));label(picb,"$X$",X1,(1,-1));label(picb,"$Y$",Y1,(-1,0));label(picb,"$P'$",P1,(-1,-1));label(picb,"$O$",O,(-1,-1)); label(picb,"(b)",O+(0,-13),(0,0)); add(pica); add(shift(30*right)*picb); [/asy]

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如果$A$和$B$是给定圆上的固定点，并且$XY$是同一圆的可变直径，则确定线$AX$和$BY$的交点的轨迹。您可以假设 $AB$ 不是直径。

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[asy] 大小(300);默认笔(字体大小(8))；实数r=10；图片 pica、picb；对 A=r*expi(5*pi/6)、B=r*expi(pi/6)、X=r*expi(pi/3)、X1=r*expi(-pi/12)、Y=r*expi(4*pi/3)、Y1=r*expi(11*pi/12)、O=(0,0)、P、P1； P = 扩展名(A,X,B,Y);P1 = 扩展名(A,X1,B,Y1);路径 circ1 = 圆((0,0),r);画(异食癖，circ1);画(异食癖，B--A--P--Y--X);点(异食癖，P^^O);标签(异食癖,"$A$",A,(-1,1));标签(异食癖,"$B$",B,(1,0));标签(异食癖,"$X$",X,(0,1));标签(异食癖,"$Y$", Y,(0,-1));标签(异食癖,"$P$",P,(1,1));标签(异食癖,"$O$",O,(-1,1));标签(异食癖,"(a)",O+(0,-13),(0,0));绘制(picb，circ1)；绘制(picb，B--A--X1--Y1--B)；点(picb，P1^^O)；标签(picb,"$A$",A,(-1,1));标签(picb,"$B$",B,(1,1));标签(picb,"$X$",X1,(1,-1));la贝尔(picb,"$Y$",Y1,(-1,0));标签(picb,"$P'$",P1,(-1,-1));标签(picb,"$O$",O,(-1,-1));标签(picb,"(b)",O+(0,-13),(0,0));添加(异食癖)；添加(移位(30*右)*picb); [/asy]

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