What intrigued me about this problem was the fact that although we were
dealing with a three dimensional solution to the problem, the two
dimensional drawing in Figure 1 made the problem appear solvable in the
plane. This led to the investigation of the case in which the point is in the plane of . More precisely,
given a quadrangle and a point in its
plane, construct a line such that there is a
parallelogram which is copolar with from the point and coaxial from the
line . See Figure 2.
If is in the plane of , the construction is as follows. Join with the vertices and the exterior diagonal points and . On the line pick a point and construct lines and parallel to and , respectively. meets in a point and meets in a point . Join the points and . The line meets at and at . Join the points and which meet at . Join and . This line intersects at and the figure is the required parallelogram.
Proof. Triangles and are coaxial from the line since meets in , meets in and meets in . By Desargues' Theorem, these triangles are copolar. By construction, is parallel to . Therefore, the pole is an ideal point, and thus is parallel to . Since triangles and are coaxial from line , we see, by a similar argument, that is parallel to . All that remains to prove is that and are concurrent at .
Suppose meets at . Triangles and are coaxial from the line , and therefore they
must be copolar. Line was constructed parallel to ; therefore is parallel to . On the other hand, it has already been established that is parallel to . Since one and only
one line can be drawn parallel to a given line through a given point, is on line . Therefore, lines
are concurrent on .
It is worth noting that if is chosen to be on the circle with diameter , the parallelogram will be a rectangle.
Moreover, it appears that there exist points on the circle for which the rectangle becomes a square. Below, in Figure 4, we see rectangles that are produced by choices of close to and close to . In the drawing on the left ; whereas, on the right . As moves from the position on the left to the position on the right, the dimensions of the rectangle change continuously from to . In Figure 5 we see near a point that produces a square.
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The translation was initiated by Bill Richardson on 2007-04-16