APMO 2001 - Asian Pacific Mathematics Olympiad Question 5

Find the greatest integer $n$ such that there are $n+4$ points $A,B,C,D,X_1,\ldots,X_n$ in the plane with $AB\ne CD$ and, for each $i=1,\ldots,n$, triangles $ABX_i$ and $CDX_i$ are congruent.