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 3 of 6: Pair the possible intersections
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AB<CD  ⟹  the two intersection points from one paired circle configuration cannot coexist with those from the opposite pair.AB<CD\implies\text{the two intersection points from one paired circle configuration cannot coexist with those from the opposite pair}.
Detailed analysis

The official argument considers, for example, two points from the circles centered at B and C. Congruence makes both points lie on the perpendicular bisector of AD, while their joining line is perpendicular to BC; hence AD is parallel to BC. If AB<CD, the opposite circle pair then cannot supply congruent triangles: on the relevant perpendicular bisector the required two distances have the wrong order. The case AB>CD is symmetric.