triangle, parallelograms, incidence --- hints

Dick Lane
Use the following steps to construct a figure. 1) Pick non-collinear points A, B, C [these can be moved] 2) Pick point M inside triangle A.B.C [point M can be moved] 3) Locate point M_c such that quadrilateral A.M.B.M_c is a parallelogram. 4) Locate point M_b such that quadrilateral A.M.C.M_b is a parallelogram. 5) Locate point M_a such that quadrilateral B.M.C.M_a is a parallelogram.
Examine line segments A.M_a, B.M_b, C.M_c --- are they related in any special fashion? The Generalized Conception in Problem Solving, A. V. Brushlinskii Analyses of Reasoning Processes, pages 45--55