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similar isosceles on sides of convex ABCD, square condition, rhombus wanted

Source: Romania TST 1995 4.4

February 17, 2020
geometryrhombussquareisosceles

Problem Statement

Let ABCDABCD be a convex quadrilateral. Suppose that similar isosceles triangles APB,BQC,CRD,DSAAPB, BQC, CRD, DSA with the bases on the sides of ABCDABCD are constructed in the exterior of the quadrilateral such that PQRSPQRS is a rectangle but not a square. Show that ABCDABCD is a rhombus.
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