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Construction of parallelogram in space

Source: Czech and Slovak Olympiad 1956, National Round, Problem 2

August 16, 2022
geometry3D geometryconstructive geometryconstructionparallelogramnational olympiad

Problem Statement

In a given plane ϱ\varrho consider a convex quadrilateral ABCDABCD and denote E=AC∩BD.E=AC\cap BD. Moreover, consider a point V∉ϱV\notin\varrho. On rays VA,VB,VC,VDVA,VB,VC,VD find points A′,B′,C′,D′A',B',C',D' respectively such that E,A′,B′,C′,D′E,A',B',C',D' are coplanar and A′B′C′D′A'B'C'D' is a parallelogram. Discuss conditions of solvability.
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