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.[asy]
size(200);
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+"′", P, Q);
label(""+lbl+"",D2+rotate(theta)*(r*P), rotate(theta)*Q);
}[/asy] AMCUSA(J)MOUSAMOgeometrygeometric transformationhomothetyratio