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 1 of 6: Extract the circle loci
Previous stepNext step
{AXi,BXi}∩{CXi,DXi} contains the side lengths CD,AB.\{AX_i,BX_i\}\cap\{CX_i,DX_i\}\text{ contains the side lengths }CD,AB.
Detailed analysis

Since AB and CD are unequal but the two triangles are congruent, CD must match one of AX_i or BX_i, while AB must match one of CX_i or DX_i. Thus each X_i lies on one of two circles centered at A or B with radius CD and one of two circles centered at C or D with radius AB.