Problem 5

Find the greatest integer

n

for which there are

n+4

points

A,B,C,D,X_1,\dots,X_n

in the plane with

AB\ne CD

such that, for each

i=1,2,\dots,n

, triangles

ABX_i

and

CDX_i

are congruent.

Step 6 of 6: Use the four witnesses

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X'=XY\cap AC,\qquad Y'=XY\cap BD,\qquad \{X,Y,X',Y'\}.

Detailed analysis

The cyclic equal-arc construction gives AB parallel to CD and the equal-angle relations stated in the official solution. Consequently each of X, Y, X', and Y' has the same three side lengths in triangles AB(point) and CD(point). Thus the four triangles are congruent in pairs, AB is not equal to CD, and n=4 is attained.