In Memoriam Larry Evans who died on May, 22 2004
Larry Evans has found a conic passing through X(13), X(14) [Fermat points], X(15), X(16) [isodynamic points], X(17), X(18) [Napoleon points]. Hence, it cannot exist a non-degenerate circum-cubic through these six points.
The Neuberg cubic K001 and the Brocard (second) cubic K018 are both circular circum-cubics containing X(13, X(14), X(15), X(16). They generate a pencil of K0 cubics through these nine points we call the Evans pencil. The singular focus of each cubic lies on the Parry circle, centered at X(351), passing through X(2), X(15), X(16), X(23), X(110), X(111), X(352), X(353)...
These cubics are all orthopivotal cubics O(P) with orthopivot P on the Brocard axis (see "Orthocorrespondence and Orthopivotal Cubics" in the Downloads page). The real infinite point on the cubic is that of the line GP. The singular focus F is F_P(P). If F1 and F2 are the foci of the Steiner in-ellipse, F_P(P) is the inverse in the circle with diameter F1F2 of the reflection of P in the line F1F2.
Now, if P and Q are harmonic conjugates with respect to O and K, the cubics O(P) and O(Q) are isogonal conjugates. Here is a list of i,j for which X(i), X(j) are such points : 15,16 — 32,39 — 50,566 — 52,569 — 58,386 — 61,62 — 187,574 — 216,577 — 284,579 — 371,372 — 389,578 — 500,582 — 567,568 — 570,571 — 572,573 — 575,576 — 580,581 — 583,584 — 1151,1152. Obviously, O(X3) = K001 and O(X6) = K018 are self-isogonal conjugates.
The Evans pencil contains :
— three pK namely :
— three nK0 namely :
— five focal cubics namely :
— a number of other cubics containing several remarkable points.
The following table shows a selection of these cubics where P is the orthopivot, F the singular focus and Inf the real point at infinity. The red point is the "last" common point with the circumcircle.