MathLabs

Problem 5

Find the greatest integer nn for which there are n+4n+4 points A,B,C,D,X1,…,XnA,B,C,D,X_1,\dots,X_n in the plane with AB≠CDAB\ne CD such that, for each i=1,2,…,ni=1,2,\dots,n, triangles ABXiABX_i and CDXiCDX_i are congruent.
Step 5 of 6: Construct four points
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A=e0i, B=e30∘i, Y=e120∘i, C=e150∘i, D=e240∘i, X=e270∘i.A=e^{0i},\ B=e^{30^\circ i},\ Y=e^{120^\circ i},\ C=e^{150^\circ i},\ D=e^{240^\circ i},\ X=e^{270^\circ i}.
Detailed analysis

Take six points on one circle in the cyclic order A,B,Y,C,D,X, with the indicated central angles. Then AYD is equilateral and the arcs AB, YC, DX are equal and less than 60 degrees, exactly the official construction.