A Known Problem of Geometry and Its Cases of Indetermination

A recent note induced me to resume the classic problem (Po) and its natural generalization: (P0) To construct a square, the sides of which go through four given points. (P) To construct a quadrilateral Q(XYZT) similar to a given quadrilateral q with its sides XY, YZ, • • • , respectively passing through four given points K, L, M, N. The solution of (P0) is classic: we immediately get a locus for each vertex, namely the circumference having KL, LM and so on for its diameter. Moreover, the diagonal XZ of the required square must again cut the half circumference of diameter NK at its middle point a and, similarly, the half circumference of diameter LM &t its middle point y. This generally determines it, whence the required construction directly results. The solution extends automatically to (P) : (i) We know a circular locus for each vertex of the unknown quadrilateral. (ii) We also know the two points a, 7 where the diagonal XZ intersects the loci of the vertices X, Z. This generally allows us to draw the required quadrilateral and : (iii) The quadrilateral Q constructed in that way is actually similar to q and, therefore, satisfies all the conditions of the problem. The verification of the latter fact offers no difficulty; but it must be insisted on, for it is quite essential for what we are going to say. Indeed t instead of reasoning on the diagonal XZt we could operate in the same way on the other diagonal YT, of which, similarly, we know two points j3, 5, respectively belonging to the circumferences which are the loci of the vertices Y and T\ and it results from (iii) that this second construction necessarily gives the same result as the first one. But, moreover, a singular case may occur, namely that in which