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Triangles with vertices on sides of square

Source: Czech and Slovak Olympiad 1966, National Round, Problem 3

June 29, 2024
geometryTrianglesquare

Problem Statement

A square ABCD,AB=s=1ABCD,AB=s=1 is given in the plane with its center SS. Furthermore, points E,FE,F are given on the rays opposite to CB,DACB,DA, respectively, CE=a,DF=bCE=a,DF=b. Determine all triangles XYZXYZ such that X,Y,ZX,Y,Z lie in this order on segments CD,AD,BCCD,AD,BC and E,S,FE,S,F lie on lines XY,YZ,ZXXY,YZ,ZX respectively. Discuss conditions of solvability in terms of a,b,sa,b,s and unknown x=CXx=CX.
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