Rigby's Eight Point Conic for a Quadrilateral

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.

Rigby's Eight Point Conic for a Quadrilateral
Though Michael Fox & I had a (different) proof for the cyclic quadrilateral case for the result described above, 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:
a) the conic 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. ∠PEA = ∠PGD; ∠PFB = ∠PHA)
b) and more generally, the conic 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. ∠EPB = ∠CPG; ∠FPC = ∠DPH).

The 1st sketch directly below shows a cyclic quadrilateral ABCD. Click on the 'Link to' buttons in the bottom right corner of the dynamic sketch below to navigate to the general or orthodiagonal quadrilateral cases.

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

Explore
1) In the 1st sketch above for the cyclic quadrilateral ABCD, drag any of the vertices until ∠APB between the diagonals becomes more or less 90°. What do you notice regarding the conic? Check your conjecture by changing the shape of ABCD & further dragging.
2) In the 2nd sketch above for the general quadrilateral ABCD, drag any of the vertices until ∠APB between the diagonals becomes more or less 90°. What do you notice regarding the conic now (compared to 1))? Can you drag E and F so that the conic becomes a circle? Check your conjecture by changing the shape of ABCD & further dragging. Compare with 3) below.
3) Navigate to the 3rd sketch for the orthodiagonal quadrilateral ABCD by clicking the 'Link to orthodiagonal quad' button. What do you notice about the conic now? Check your observations by dragging - also check concave & crossed cases of ABCD - do your observations still hold?

Challenge
4) Can you prove all your observations above? Can you prove the results in more than one way?
(Compare your proofs with those given further down).

Note: The general quadrilateral 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.

Some Proofs
i) John Rigby's Solution is available on JSTOR at: Problem Corner or directly from Rigby Eight Point Conic General Proof.
[Comment: Rigby's proof which not only verifies and explains the result in terms of projective geometry, also illustrates quite nicely the so-called discovery function of proof, by showing how it generalizes to equi-inclined lines (as shown in the 1st dynamic sketch above), and to a general quadrilateral (as shown in the 2nd, linked, dynamic sketch above).]
ii) A proof for the 8-point circle for a (convex) orthodiagonal quadrilateral case is given in Josefsson (2012). It is left to the reader to adapt or modify the proof for the concave and crossed cases.
iii) After posting the 8-point circle problem on Facebook in Sept 2026, the following two proofs for the 8-point circle for a (convex) orthodiagonal quadrilateral case were respectively received from Jean-Pol Coulon and Ichung Chen: Eight Point Circle Orthodiagonal Quad Solutions.

References
Fox, M. & De Villiers, M. (2005). The 8pt Conic Problem. (Unpublished proof of special cyclic case formulated for ⊥'s right at the top).
Josefsson, M. (2012). Characterizations of Orthodiagonal Quadrilaterals. Forum Geometricorum, Vol 12, 13–25. (See Theorem 4, pp. 16-17).
N.J.L. (2007). Problem Corner. The Mathematical Gazette, 91:521, 358-362.

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.

Another 8-point circle of orthodiagonal quad
Here is another interesting, but relatively easier to prove 8-point circle associated with an orthodiagonal quadrilateral:
Eight-Point Circle (Pamfilos)
Eight-Point Circle Theorem (Wolfram MathWorld)
Eight-Point Circle (Cut The Knot)
Eight Point Circle Theorem (Go Geometry)

More Equi-inclined lines
Other examples of results involving equi-inclined lines:
Equi-inclined Lines Problem
Equi-inclined Lines Generalizations of Viviani's Theorem
A generalization of Neuberg's Theorem and the Simson line
A variation of Miquel and its generalization
Equi-inclined Lines Problem
Generalizations of a theorem by Wares
The Equi-inclined Bisectors of a Cyclic Quadrilateral
The Equi-inclined Lines to the Angle Bisectors of a Tangential Quadrilateral

External Links
John Rigby (mathematician) (Wikipedia)
Orthodiagonal quadrilateral (Wikipedia)
Martin Josefsson (Academia)
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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Created by Michael de Villiers, 9 March 2008 with Cinderella 2, updated to WebSketchpad, 9 May 2021; updated 18/23 Sept 2026.