P for Parallelogram
Point is located inside a convex quadrilateral. Is it possible to inscribe in the quadrilateral a parallelogram such that is on one of its sides?
The parallelogram is considered to be inscribed if all of its vertices lay on the sides of the quadrilateral.
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1 solution
(1) If the point P is not at the intersection of diagonals of the quadrilateral, run a line parallel to a diagonal it is not on. The line will intersect two adjacent sides of the quadrilateral at points Q and R. Lines through these points parallel to the other diagonal then give the remaining vertices of the parallelogram. The vertices of this parallelogram will each be on a different side of the original quadrilateral.
(2) If the point P is at the intersection of the diagonals, run an arbitrary line through it. It will intersect two opposing sides at points Q and R.
(a) If the sides are not parallel, draw lines through points Q and R parallel to the opposing sides. One of them will intersect the quadrilateral ABCD at a point S such that a line through S parallel to QR will finish the parallelogram. Two of the vertices of the parallelogram, Q and T in the image above, will be on the same side of the quadrilateral ABCD.
(b) If the sides are parallel, any line parallel to QR and intersecting them will finish the parallelogram. Each of the sides will have a pair of vertices on it, however.