Nine-point conics, cyclocevian conjugates and related topics

Most of what follows is probably well-known. A web search found fragments, but no single source
appears to contain all of the facts we list. After it was written, I discovered Bernard Gibert's paper
on the subject. There is considerable overlap. The paper is reference [BG].

We use many standard notions, such as tripolar, crosspoint, cycclocevian conjugate. These are
defined in [ETC-G], and many other places. We refer to points in the notation of [ETC], so that
the centroid is X(2), the circumcentre X(3), the orthocentre X(4), the nine-point centre X(5).

Elementary proofs of the early results are available here.

Non-standard definitions and notation

For a point P,

C(P) denotes the circumconic with perspector P, this is a projective notion,

T(P) denotes the tripolar of P, it is the polar of P with respect to C(P), so is a projective notion,

The traces of P are the intersections of the cevians of P with the sidelines, these are undefined
if P lies on a sideline.

if X and Y are points, not both on T(P), the P-harmonic of XY is the point Z such that (X,Y,W,Z)
is harmonic, where W = T(P)nXY. Note that, for finite X,Y, the G-harmonic is the mid-point of XY.

We also introduce the idea of the P-complement of U, which has simple geometry.
If P = p:q:r, U = u:v:w, the P-complement of U has barycentrics p(v/q+w/r): . : .
We also have the P-anticomplement of U with barycentrics p(-u/p+v/q+w/r): . : .
These are inverse projective mappings. They can be recovered from the familiar operations
of crosspoint, cevapoint, ceva and cross-conjugates, and P2-reciprocal conjugation,
but are really more fundamental operations.

The centre of a conic is the pole of T(G) in the conic. We generalize this as follows.
For conic C, the P-centre of C is the pole of T(P) in C.
When C is the circumconic C(Z), the P-centre is the P-Ceva conjugate of Z.
We note the following simple result, which justifies the use of "conjugate" in this context.

Lemma
If W is the P-Ceva conjugate of Z, then Z is the P-Ceva conjugate of W

Generalization of the nine-point circle

Part (1) of the following result is mentioned in [Y] as "well-known". See also [OW].
When U is X(2), parts (1) and (2) are equivalent to the nine-point conic result in Mathworld [MW] .

Theorem 1 The eighteen-point conic
Suppose that U = [u,v,w] and P = [p,q,r] are any points.
(1) There is a conic C(U,P) passing through the (six) traces of U and P.
(2) The conic C(U,P) contains the U-harmonics of AP, BP and CP.
(3) The conic C(U,P) contains the P-harmonics of AU, BU and CU.
Let Q be the crosspoint of U and P.
(4) The intersections of C(Q) and C(U,P) lie on T(U)uT(P).
C(P,U) is the P-complement of C(Q), and the U-complement of C(Q).
Let U', U" be the poles of T(U) in C(U,P), C(Q), (i.e. U-centres) respectively, and
let P', P" be the poles of T(P) in C(U,P), C(Q), (i.e. P-centres) respectively.
See Notes on this and others below.
(5) U', U", P',P" lie on UP, with (UPU'U") and (UPP'P") harmonic.
Let U"', P"' be the intersections of UP with T(U), T(P), respectively.
(6) (PU"U'U"') and (UP"P'P"') are harmonic.
Let R be the intersection of T(U) and T(P).
(7) The polars of P and U in C(U,P) and in C(Q) concur at R, so R is the pole of PU in either conic.
(8) The conic C(Q) meets C(U) at the U2-isoconjugate of R, and C(P) at the P2-conjugate of R.
Let S be the point [(1/v+1/w)(1/q+1/r),(1/u+1/w)(1/p+1/r),(1/u+1/v)(1/p+1/q)].
This is the barycentric product of the complements of the isotomic conjugates of U and P.
Let T be the barycentric product of U and P.
(9) C(U,P) and C(S) are have the same intersections with T(T) and T(X(2)) : x+y+z=0.
Let K = (u+v)(v+w)(w+u)(p+q)(q+r)(r+p)/2(uq+vp)(vr+wq)(wp+ur).
(10) If K > 0, C(U,P) and C(S) are homothetic, with ratio K½
(11) If K < 0, C(U,P) and C(S) are hyperbolas with the same asymptotes, but in opposite quadrants.
(12) If K = 0, C(S) degenerates into a sideline and another line.
This occurs if U or P lies on a line through a vertex parallel to the opposite side.
(13) If 1/K = 0, C(U,P) degenerates. This occurs when P (resp. U) lies on the line through a vertex
and the intersection of T(U) (resp. T(P)) with the opposite side. This line is then part of C(U,P).
(14) When U or P lies on a sideline, C(U,P) and C(S) both degenerate.

Notes
The first sixteen points are those in parts (1)-(4). The other points appear in Theorem 2 below.
Those in (4) may be imaginary - as they are for the nine-point circle C(G,H).
The nine points in in parts (1) and (2) are well-documented.
The intersections with T(U) complete the eleven cited in [11].

(4) The result means that C(Q) is the circumconic in the pencil containing C(P,U) and T(U)uT(P).

(4,5,6) coordinates
Q = (1/vr+1/wq,1/wp+1/ur,1/uq+1/vp) - the crosspoint of P and U
U" = [(u(q/v+r/w)),..,..] the U-complement of P
U' = [u(2p/u+q/v+r/w),..,..] the U-complement of U"
P" = [p(v/q+w/r)),..,..] the P-complement of U
P' = [p(2u/p+v/q+w/r),..,..] the P-complement of P".
U"' = [u(-2p/u+q/v+r/w,..,..]
P"' = [p(-2u/p+v/q+w/r,..,..]

(5,6) The harmonic relationships lead to a nice result
Corollary
U' is the U-harmonic of U and the U-harmonic of UP,
P' is the P-harmonic of P and the P-harmonic of UP.

(8) R = (1/vr-1/wq,1/wp-1/ur,1/uq-1/vp).
The result allows us to identify Q as the perspector of the circumconic through these isoconjugates.
The isoconjugates are the trilinear poles of UQ and PQ.
(9) This means that C(U,P) is a circle if and only if S = X(6). The definition in [ETC-G]
shows that U and P must be cyclocevian conjugates. The formula for S then shows that
P is
the isotomic conjugate of
the anticomplement of
the isogonal conjugate of
the complement of
the isotomic conjugate of U.
See Darij Grinberg's Theorem 9 in [DG].

Further results on conics

It is well-known that the nine-point circle is the locus of centres of circumconics through X(4).
Parts (1) and (2) of the following Theorem are easy consequences of the above Lemma.

Theorem 2
We use the notation of Theorem 1.
(1) C(U,P) is the locus of P-centres of circumconics through U.
Equivalently, C(U,P) is the image of T(U) under P-Ceva conjugation.
(2) C(U,P) is the locus of U-centres of circumconics through P.
Equivalently, C(U,P) is the image of T(P) under U-Ceva conjugation.
R is the intersection of the T(U) and T(P) - the perspector of the circumconic through U and P
Let V, W be the U- and P-Ceva conjugates of R.
(3) V and W lie on C(U,P).
(4) V is the P-centre of the circumconic through U and Q (and U" and U+ - see Theorem 4).
V is the U-centre of the circumconic through U and P.
(5) W is the U-centre of the circumconic through P and Q (and P" and P+.
W is the P-centre of the circumconic through U and P.
(6) V, W lie on T(R).
(7) V lies on the inconic with perspector P,
(8) W lies on the inconic with perspector U.

Notes
(3) R is the perspector of the circumconic through U and P, so (3) follows from (1) and (2).
These points V and W have appeared before. See [DG2] for a discussion and a geometrical construction.
If U and P are triangle centres, so are V and W - we now have two centres on C(U,P)
V = [(1/p) (1/vr-1/wq)2,..,..]
W = [(1/u) (1/vr-1/wq)2,..,..]

A result relating C(U,P) and C(Q)

We know that the nine-point circle bisects a segment joining X(4) to a point on the circumcircle.

Theorem 3
We use the notation of Theorem 1, but now assume that P is not on T(U), nor U on T(P).
(1) Any segment joining U to a point on C(Q) is cut harmonically by C(U,P) and T(P).
(2) Any segment joining P to a point on C(Q) is cut harmonically by C(U,P) and T(U).

Notes
The necessary calculations were done in Maple.
U is on T(P) if and only if U is on C(Q), and similarly for P.

Eight more points

Let us refer to the points V, W above as the points V1, W1.
Bernard Gibert has added two new points as follows :

V2 = U-centre of the circumconic through P touching PU at P,
W2 = P-centre of the circumconic through U touching PU at U.

With the idea that a point on C(U,P) is defined by the U-Ceva conjugate of a point on T(P),
we obtain more points by intersecting T(U) and T(P) with lines defined by U, P and Q.

V3 = U-Ceva conjugate of intersection of UP and T(P),
W3 = P-Ceva conjugate of intersection of UP and T(U),
V4 = U-Ceva conjugate of intersection of PQ and T(P),
W4 = P-Ceva conjugate of intersection of UQ and T(U),
V5 = U-Ceva conjugate of intersection of UQ and T(P),
W5 = P-Ceva conjugate of intersection of PQ and T(U).

Note that these points can be constructed as Ceva conjugates of line intersections,
apart perhaps for V2, W2. The barycentrics are uniformly nasty.

We do have some nice geometry :

V1.W4, W2.V3, W1.V5 meet at U,
W1.V4, V2.W3, V1.W5 meet at P,
W1.W3, W2.W5, V1.V5, meet at U', the U-centre of C(U,P).
V1.V3, V2.V5, W1.W5, meet at P', the P-centre of C(U,P),
V1.W3, W1.V3 meet at Q,
W1.V2, V1.W2 meet at the R2, the cevapoint of V1, W1,
V1.W1, V2.W2, U.P meet at R3, the V1-Hirst inverse of W1.

Thus
W2 is the intersection of U.V3 and U'.W5, which are constructible,
V2 is the intersection of P.W3 and P'.V5, which is also constructible.

V2 is the U-Ceva conjugate of the P-isoconjugate of the tripole of UP,
W2 is the P-Ceva conjugate of the U-isoconjugate of the tripole of UP.

Prasolov points

In the notes on X(68), Kimberling refers to Prasolov's book containing the original result
that the antipodes on the nine-point circle of the traces of X(4) give a triangle perspective
with ΔABC at X(68).

Definition
If P is a point not on a conic C, and Q is a point on C, then the P-antipode of Q on C is the
second intersection of PQ with C.
If ΔXYZ is a triangle with vertices on C, then the P-antipodal triangle of ΔXYZ is the triangle
with vertices the P-antipodes of X, Y, Z on C.

Theorem 4
We use the notation of Theorem 1.
(1) The P'-antipodal triangle of the cevian triangle of P is perspective with ΔABC at U.
The P'-antipodal triangle of the cevian triangle of U is perspective with ΔABC at a point P+.
P+ lies on QP'.
(2) The U'-antipodal triangle of the cevian triangle of U is perspective with ΔABC at P.
The U'-antipodal triangle of the cevian triangle of P is perspective with ΔABC at a point U+.
U+ lies on QU'.

Notes
We call the points P+, U+ the Prasolov points of P, U.
P+ = [p2/(p2vw-u(uqr+vrp+wpq)),..,..].
U+ = [u2/(u2qr-p(pvw+qwu+ruv)),..,..].

The first clause of each part allows us to construct the intersections of the conic with the cevians
of U and P. For example, the line joining a vertex of the cevian triangle of U to the point U' meets
the corresponding cevian of P at a point of the conic.

Maple shows that the nine-point conic for P and P+ meets that for P and U
at W, and similarly for U and U+. See also Theorem 2 (7),(8).

The case U = G = X(2)

C(G,P) contains
the G-harmonics (mid-points) of AB,BC,CA, and of AP,BP,CP.
the P-harmonics of AB,BC,CA, and of AG, BG,CG.
These are the twelve basic points on C(G,P).
Q = S, and each is the complement of the isotomic conjugate of P.
C(G,P) meets C(Q) on T(P) and on T(G), the Line at Infinity.
This gives another four points on C(G,P), they may not be real.
C(G,P) is the complement of C(Q).
G' and G" are the centres of C(G,P) and C(Q) respectively.
G" is the complement of P.
G' is the complement of G", and hence the mid-point of PG".
(GPG'G") is harmonic, so now PG' = 3G'G
G' is the mid-point of G and the mid-point of PG (generalised below *)
P' and P" are the poles of T(P) in C(G,P) and C(Q) respectively.
Then P', P" lie on GP (as do G', G" of course).
(PGP'P") is harmonic, so
(*) P' is the P-harmonic of P and the P-harmonic of P and G.
P' is the P-harmonic of GP".
The polars of G and P in C(G,P) and C(Q) are parallel to T(P).
C(G,P) is the locus of mid-points of PZ, for Z on C(Q).
Thus C(G,P) and C(Q) are homothetic, centre P, factor 2.
C(G,P) is the locus of centres of circumconics through P.
C(G,P) is the locus of poles of T(P) in circumconics through G.
C(G,P) contains V, the centre of the circumconic through G and P.
This lies on the inconic with perspector P.
C(G,P) contains W, the centre of the circumconic through Q and P.
This lies on the inconic with perspector G - the Steiner Inellipse.
These points V and W complete the eighteen points on C(G,P).
The lines joining the G-harmonic of AB to that of CP (and similar) meet at G'.
The lines joining the P-harmonic of AB to that of CG (and similar) meet at P'.

The case U = X(2), P = X(4)

Q = X(6), the crosspoint of X(2) and X(4) is X(6), so C(Q) is the circumcircle and C(X(2),X(4)) is a circle.
It passes through the mid-points of the sides, AX(4), BX(4), CX(4) and the feet of the altitudes.
C(X(2)X(4)) is the nine-point circle.
By Theorem 1(3), the second intersections with the medians are X(4)-harmonics of AX(2), BX(2),CX(2).
X(2)X(4) is the euler line.
T(X(2)) is the line at infinity, T(X(4)) is the orthic axis.
The tripolars and poles concur at
R =X(523), the infinite point on the orthic axis.
The circumcircle passes through the isoconjugates of R. These are X(99) and X(107).
X(2)' = X(5), the centre of the nine-point circle
X(2)" = X(3), the centre of the circumcircle
X(2)"' = X(30), the infinite point on the euler line
X(4)' = X(427), the pole of the orthic axis in the nine-point circle
X(4)" = X(25), the pole of the orthic axis in the circumcircle
X(4)"' = X(468). the intersection of the euler line and the orthic axis.
We have the known harmonics (X(2)X(4)X(3)X(5)), (X(3)X(4)X(5)X(30)), (X(2)X(25)X(427)X(468)).
V1= X(115), the centre of the Kiepert hyperbola (perspector R), and the X(4)-centre of the circumconic
through X(2) and X(6). The latter has perspector X(512)), on T(X(2)).
W1 is X(125), the centre of the Jerabek hyperbola, and the X(4)-centre of the Kiepert hyperbola.
The Jerabek hyperbola has perspector X(647), on the orthic axis. It passes through X(4) and X(6).
V1 = X(115), the X(2)-Ceva conjugate of X(523), the meet of T(X(2)) and T(X(4)),
W1 = X(125), the X(4)-Ceva conjugate of X(523), the meet of T(X(4)) and T(X(2)),
V2 = X(136), the X(2)-Ceva conjugate of X(2501), the X(4)-isoconjugate of the triople of X(2).X(4)
W2 = X(127), the X(4)-Ceva conjugate of X(525), the X(2)-isoconjugate of the triople of X(2).X(4)
V3 = X(1560), the X(2)-Ceva conjugate of X(468) the meet of X(2).X(4) and T(X(4),
W3 = X(113), the X(4)-Ceva conjugate of X(30), the meet of X(2).X(4) and T(X(2)),
V4 = X(133), the X(2)-Ceva conjugate of X(1990), the meet of X(4).X(6) and T(X(4)),
W4 = X(126), the X(4)-Ceva conjugate of X(524), the meet of X(2).X(6) and T(X(2)),
V5 = X(114), the X(2)-Ceva conjugate of X(230), the meet of X(2).X(6) and T(X(4)),
W5 = X(132), the X(4)-Ceva conjugate of X(1503), the meet of X(4).X(6) and T(X(2)).
P+ = X(68).
U+ is unlisted, it has barycentrics [1/(a2-b2-c2)(a4-b4-c4),..,..].

The case U = X(1), P = X(57)

Q = X(6), so C(Q) is the circumcircle,
R = X(513), so T(U) and T(U) are parallel, they are perpendicular to UP,
U' = X(354),
U" =X(55),
U"' = X(1155),
P' = X(65),
P" = X(56),
P"' = X(1319),
V = X(244),
W = X(2170),
U+ is unlisted it has barycentrics [a(a-b-c)/(a2+b2+c2-2ab-2ac),..,..],
P+ = X(34).

The X(65)-antipodal triangle of X(1) on the circumconic of the cevian triangles of X(1) and X(57)
is perspective with ΔABC at X(34), which lies on X(6)X(65) ( = X(6)X(19) ).

Notes on the calculations

The calculations were done by Maple.
The results (10), (11) of Theorem 1 need some explanation.
We look at equations for C(U,P) and C(S).
We replace the variable x by X-y-z, so the line at infinity becomes X = 0.
In each case we replace y by y+mX, and z by z+nX to eliminate terms in yX and zX.
The equations now have the forms F(y,z) + gX2 and F(y,z) + hX2.
The stated value of K is h/g.
If K > 0, the conics are plainly homothetic as stated.
Both conics contain real points, so we cannot have K < 0 unless we have hyperbolas
related as stated.

Further notes

We know that, if X is on the circumconic C(Q), then there are four related points on C(U,P).

(1) the U-comp of X, and
(2) the P-harm of U,X.
(3) the P-comp of X, and
(4), the U-harm of P,X.

(1),(2) lie on UX, and (3),(4) lie on PX, so we have found the points on C(U,P) on these lines.

There are two obvious points on C(Q), the final intersections with the circumconics C(U) and C(P).
This yields eight points on C(U,P), but only two are new. These are

V6 = U-comp of C(P)nC(Q),
W6 = P-comp of C(U)nC(Q).

We have some additional concurrences :

V6 is the intersection of U.W5, and U'.V4,
W6 is the intersection of P.V5 and P'.W4.

V6 is the U-Ceva conjugate of the P-isoconjugate of T(U)nT(P),
W6 is the P-Ceva conjugate of the U-isoconjugate of T(U)nT(P).

The case U = X(2), P = X(4).

V1 = X(115),   W1 = X(125),
V2 = X(136),   W2 = X(127),
V3 = X(1560), W3 = X(113),
V4 = X(133),   W4 = X(126),
V5 = X(114),   W5 = X(132),
V6 = X(122),   W6 not in ETC.

Of course, W6 = X(4)X(114)nX(427)X(126).

Antipodes

Some of these are special cases of a more general theory of antipodal points.

X,Y on C(U,P) are Z-antipodes if Z is on XY.
U', P' are the U- and P-centres of C(U,P).
I(U), I(P) are the inconics with perspectors U,P.

From the above, we have

Theorem A
For X on C(U,P),
the U-antipode of X is the P-harm of U and the U-anti of X,
the P-antipode of X is the U-harm of P and the P-anti of X.

We also have the following, proved by Maple computation.

Theorem B
The points X,Y on C(U,P) are U'-antipodes if and only if either
(a) the P-Ceva conjugates of X,Y (on T(U)) are conjugate in C(U,P), or
(b) the U-Ceva conjugates of X,Y (on T(P)) are conjugate in I(U).
The points X,Y on C(U,P) are P'-antipodes if and only if either
(c) the U-Ceva conjugates of X,Y (on T(P)) are conjugate in C(U,P), or
(d) the P-Ceva conjugates of X,Y (on T(U)) are conjugate in I(P).

All of these are special cases of a theorem in antipodal points once we note that C(X,X) = I(X).

The results (b) and (d) clearly suggest two further cases. First, we need to describe
two further points associated with C(U,P).

Lemma
QU = U-comp Q is the perspector of C(U,P) as a circumconic of the cevian triangle of U.
QU lies on U.Q, W1.W4, W2.W3.
QP = P-comp Q is the perspector of C(U,P) as a circumconic of the cevian triangle of P.
QP lies on P.Q, V1.V4, V2.V3.

Theorem C
The points X,Y on C(U,P) are QP-antipodes if and only if
the U-Ceva conjugates of X,Y (on T(P)) are conjugate in I(P).
The points X,Y on C(U,P) are QU-antipodes if and only if
the P-Ceva conjugates of X,Y (on T(U)) are conjugate in I(U).

The case U = X(2), P = X(4).
Q = X(6),
QU = X(141),
QP = X(53).

References

[BG] Bicevian Conics

[ETC-G] Glossary for ETC

[OW] The nine-point conic theorem, Theorem 6.32
An Introduction to Projective Geometry, O'Hara and Ward (Oxford 1937), p. 135,