Eight Point Conic for Cyclic Quadrilateral

Problem
The following problem was posed by myself and Michael Fox in the Problem Corner of the Mathematical Gazette, Nov 2006: The diagonals of a cyclic quadrilateral ABCD meet at P. The feet of the perpendiculars from P to AB, BC, CD, DA are E, F, G, H. The lines EP, FP, GP, HP meet the opposite sides of ABCD at K, L, M, N. Prove that E, F, G, H, K, L, M, N lie on a conic.

Eight Point Conic for Cyclic Quadrilateral

Note
The result is also true if ABCD becomes a crossed quadrilateral, but the above dynamic sketch, due to certain construction limitations, is unfortunately not able to show it. However, as shown by the Cinderella sketch below, the result is also true for the crossed case.

Free Download
Download the dynamic geometry software Cinderella 2 for FREE from here, and use it (after unzipping) to view & manipulate the Cinderella Eight Point Conic sketch illustrating the above result.

Challenge
Can you explain why (prove that) the above result is true? Can you generalise further?
(If stuck, go to Proof & Further Generalizations).

Related Links
Water Supply: Four Towns (Rethinking Proof activity)
Triangle Altitudes (Rethinking Proof activity)
Rigby's Eight Point Conic for a Quadrilateral
Nine Point Conic and Generalization of Euler Line
Spieker Conic and generalization of Nagel line
Nine-point centre (anticentre or Euler centre) & Maltitudes of Cyclic Quadrilateral
The affine invariance of the conics
British Mathematical Olympiad problem: Conic Generalization
Parallel-Hexagon Concurrency Theorem
Merry Go Round the Triangle
Six Point Cevian Circle

External Links
Carnot's Conic Theorem (Wikipedia)
Aimssec Lesson Activities (African Institute for Mathematical Sciences Schools Enrichment Centre)
UCT Mathematics Competition Training Material
SA Mathematics Olympiad Questions and worked solutions for past South African Mathematics Olympiad papers can be found at this link.
(Note, however, that prospective users will need to register and log in to be able to view past papers and solutions.)

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