Rigby's Eight Point Conic for a Quadrilateral

Though Michael Fox & I had a (different) proof for the cyclic quadrilateral case, we were both pleasantly surprised by John Rigby's insightful solution to our problem in the Problem Corner of the Mathematical Gazette, July 2007, pp. 358-362, which showed:
1) the result still holds for a cyclic quadrilateral if the pairs of lines PE and PG, and PF and PH, are respectively EQUI-INCLINED towards the OPPOSITE SIDES of ABCD (i.e. angle PEA = angle PGD; angle PFB = angle PHA)
2) and more generally, the result generalizes even further to ANY quadrilateral ABCD if the pairs of lines PE and PG, and PF and PH, are respectively EQUI-INCLINED towards the DIAGONALS of ABCD (i.e. angle EPB = angle CPG; angle FPC = angle DPH).

Click on the Button in the bottom right corner of the dynamic sketch below to navigate to the general quadrilateral case.

Rigby's Eight Point General Conic for Cyclic & General Quadrilateral

Note: The general result is also true if ABCD becomes a concave or crossed quadrilateral, or if E or F are dragged onto the extensions of the sides, but the above dynamic sketches, due to certain construction limitations, are unfortunately not able to show it. However, you can download the dynamic geometry software Cinderella 2 for FREE from here, and use it (after unzipping) to view & manipulate the following two zipped Cinderella sketches, namely Eight Point Conic General Cyclic sketch and Eight Point Conic General Quad sketch to dynamically explore these cases.

John Rigby's Solution is available on JSTOR at: Rigby Eight Point Conic General.

Also see this webpage about a Japanese Circumscribed Quadrilateral Theorem, which is dedicated to John Rigby.

For more information on the journal go to: Mathematical Gazette.