triangle centres Nine-point circles

This is really a special case of the theory in the centre of a nine-point conic.
We now consider nine-point conics C(P,Q) homothetic with C(X6), the circumcircle.
In other words, we are looking at nine-point circles. Such C(P,Q) arise when P and
Q are cyclocevian conjugates.

As usual, for a point M,
c(M) is the complement of M,
a(M) is the anticomplement of M,
t(M) is the isotomic conjugate of M,
g(M) is the isogonal conjugate of M.
Then the cyclocevian conjugate of M is tagct(M).

From our earlier results, we have

Theorem 1
Suppose that L : Kx+Ly+Mz = 0 is a line other than the Line at Infinity.
Then C(P,Q) is a circle with centre on L if and only if
ct(P), ct(Q) are isogonal conjugates on K = nK(X6,R,t), where
R is the orthopoint of the infinite point of L, and t depends only on L.
t = 0 if and only if X3 is on L, and then K is nK0(X6,R).

The value of t is given explicitly in the Appendix to the centre of a nine-point conic.
It is linear in K,L,M, so we immediately have the

Corollary 2
For a finite point X, there are three pairs {P,Q} with C(P,Q) a circle with centre X.

Observe that, for fixed X, the relevant P,Q are associated with a pencil of cubics nK(X6,R,t)
given by the pencil of lines through X. Equally, each such pencil of cubics leads to a point X.
The members of the pencil meet in A,B,C and in three common isogonal pairs.

The equations giving the the coordinates of R and the value of t in terms of K,L,M
can be used to recover K,L,M from the point R = u:v:w (at infinity) and t. We simply
solve for K,L,M the equations giving the first two coordinates for R, and that giving t.

Theorem 3
Suppose that R is a fixed point at infinity, and that t is a constant.
If P is on nK(X6,R,t) then C(ta(P),tag(P)) is a circle with centre on a line L(R,t).
For any t, L(R,t) contains the orthopole of R.
For t = 0, L(R,t) is the polar of R in the Circumcircle.

Examination of the examples in the centre of a nine-point conic also reveals

Theorem 4
In the notation of Theorem 1,
(1) K is nK0(X6,R) if and only if X3 is on L,
(2) K is cK(#X1,R) if and only if X1 is on L,
(3) K is circular if and only if X20 is on L,
(4) K is nK(X6,R,X2) if and only if X5 is on L.

Proof Notes - using Theorem 3 and preceding remarks

K018 = nK0(X6,X523) gives L as the Euler Line; this contains X3, X5, X20; K018 is circular, contains X2.
K325 = nK0(X6,X512) gives L as the Brocard Axis: this contains X3,
K067 = nK(X6,X30,X2) gives L as the line X5-X523 (orthopoint of X30); this contains X5
K086 = cK(#X1,X514) gives L as the line X1-X7; this contains X1.
K137 = cK0(#X1,X513) gives L as the line X1-X3; this contains X1, X3
K187 = nK(X6,X525,X3) gives L as the line X20-X64; this contains X20; K187 is circular.
K248 = nK(X6,X512,X187) gives L as the line X20-X185; this contains X20; K248 is circular.
K352 = nK(X6,X522,X9) gives L as the line X8-X20; this contains X20; K352 is circular.

Note that K018, K187, K248, K352 all contain the (real) foci f1,f2 of the Steiner Inellipse.
The corresponding P, Q are then tF1, tF2, where F1,F2 are the (real) foci of the Steiner Ellipse.
This tells us that c(tF1,tF2) is a circle centre X20 (we can replace F1,F2 by the non-real foci).

Theorem 4 allows us to find the points P,Q with C(P,Q) a circle of centre X.
We simply intersect two cubics derived from lines through X. In general, if X ≠ X1,X3,
we can use X-X1 and X-X3 to get a cK and an nK0.
The case X in {X1} is special. We obtain the incircle centre X twice and an imaginary pair.
For X = X3, we could use K018 and K325 (or K137).

From earlier results, we also have

Theorem 5
The locus of points P such that P, Q = the cyclocevian conjugate of P, and the circumcentre of the
cevian triangle of P are collinear consists of
(1) the Lucas Cubic K007 = pK(X2,X69),
(2) the points P for which C(P,Q) is degenerate.
For P on K007, the cyclocevian conjugate is also on K007, and the common line also contains X20.
The points in (2) include P on the sidelines of either ΔABC or its antimedial triangle, or points for
which a vertex of the cevian triangle of Q is on a sideline of the cevian triangle of P. Equivalently,
these occur when Q is on any sideline of the anticevian triangle of P