1978 USAMO 第 2 题

$ABCD$ and $A'B'C'D'$ are square maps of the same region, drawn to different scales and superimposed as shown in the figure. Prove that there is only one point $O$ on the small map that lies directly over point $O'$ of the large map such that $O$ and $O'$ each represent the same place of the country. Also, give a Euclidean construction (straight edge and compass) for $O$.

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[asy] defaultpen(linewidth(0.7)+fontsize(10)); real theta = -100, r = 0.3; pair D2 = (0.3,0.76); string[] lbl = {'A', 'B', 'C', 'D'}; draw(unitsquare); draw(shift(D2)*rotate(theta)*scale(r)*unitsquare); for(int i = 0; i < lbl.length; ++i) { pair Q = dir(135-90*i), P = (.5,.5)+Q/2^.5; label("$"+lbl[i]+"'$", P, Q); label("$"+lbl[i]+"$",D2+rotate(theta)*(r*P), rotate(theta)*Q); }[/asy]

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$ABCD$ 和 $A'B'C'D'$ 是同一区域的方形地图，绘制为不同比例并叠加，如图所示。证明小地图上只有一个点 $O$ 直接位于大地图的点 $O'$ 上方，使得 $O$ 和 $O'$ 分别代表该国家/地区的同一地点。另外，给出一个欧几里得构造(直尺和圆规)，价格为 $O$。

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[asy] defaultpen(linewidth(0.7)+fontsize(10));实θ = -100，r = 0.3；对 D2 = (0.3,0.76);字符串[] lbl = {'A', 'B', 'C', 'D'};绘制(单位平方)；绘制(移位(D2)*旋转(θ)*缩放(r)*unitsquare)； for(int i = 0; i < lbl.length; ++i) { 对 Q = dir(135-90*i), P = (.5,.5)+Q/2^.5;标签("$"+lbl[i]+"'$", P, Q);标签("$"+lbl[i]+"$",D2+旋转(theta)*(r*P), 旋转(theta)*Q); }[/asy]

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