Results on X-logic Triangles

1. The basic definition and properties.

In our discussions, we shall consider non-degenerate triangles ΔPQR . We shall ignore cases where a vertex
lies on a sideline of ΔABC when convenient for the algebra.

Notations
We write X(n)(ΔABC) for Kimberling centre X(n) relative to ΔABC.
As usual, we write G for the centroid and H for the orthocentre of a triangle.

Definition
Given a finite point X not on a sidelines ofΔABC, and a triangle ΔPQR.
ΔABC is X-logic with ΔPQR if the lines through P parallel to AX, through Q parallel to BX and through R parallel
to CX are concurrent.
When this happens, the point of concurrence is the X-point of ΔPQR.

This generalises the notion of orthologic triangles which corresponds to our idea of H-logic triangles.

Observe that, when X is at infinity, the stated lines concur at X. This is why we include the condition "X finite".
We allow vertices of ΔPQR at infinity. The Definition implies that the triangles are X-logic for all X whenever
at two lie at infinity. This also happens when the triangles are homothetic.

Theorem 1.1
If ΔABC is X-logic with ΔPQR with X-point equal to Y,
then ΔPQR is Y-logic with ΔABC with Y-point equal to X.

This is obvious.

Theorem 1.2
Suppose that ΔPQR is a triangle as above. The points X for which ΔABC is X-logic with ΔPQR lie on a circumconic
C(ΔPQR)of ΔABC. This degenerates to the whole plane if either
(a) two vertices of ΔPQR are at infinity, or
(b) triangles ΔABC and ΔPQR are homothetic.

The algebra shows that the triangles meet the conditions for X-logicity for X at infinity or on a circumconic.
Since we have excluded cases with X at infinity, we just have the circumconic. The algebra also reveals that
the equation has the form "0 = 0" precisely when condition (a) or (b) holds.

The following key result was stated by Francois Rideaux in Hyacinthos. Proof of (1) by Maple is straightforward.
Part (2) is really just Theorem 1.1. Part (3) is another simple verification.

Lemma 1.3
Suppose that ΔABC and ΔPQR are non-homothetic triangles.
Let f denote the affine transformation taking A to P, B to Q and C to R.
(1) For X on C(ΔPQR), ΔABC is X-logic with ΔPQR with X-point on f(C(ΔPQR)).
(2) For Y on f(C(ΔPQR)), ΔPQR is Y-logic with ΔABC with Y-point on C(ΔPQR).
(3) The infinite fixed points of C(ΔPQR) are fixed by f, so also lie on f(C(ΔPQR)).

Corollary 1.3.1
If ΔABC is H(ΔABC)-logic with ΔPQR, then ΔPQR is H(ΔPQR)-logic with ΔABC.
Proof
As ΔABC is H(ΔABC)-logic with ΔPQR, H(ΔABC) lies on C(ΔPQR).
This is therefore a rectangular hyperbola. As the affine map f fixes the infinite points of the conic, the conic
f(C(ΔPQR)) is also rectangular, and hence contains H(ΔPQR). The result then follows by Lemma 1.3(2).

Corollary 1.3.2
If ΔABC is G(ΔABC)-logic with ΔPQR, then ΔPQR is G(ΔPQR)-logic with ΔABC.
Proof
As ΔABC is G(ΔABC)-logic with ΔPQR, G(ΔABC) lies on C(ΔPQR).
The affine map f maps centroids to centroids. Thus, f(C(ΔPQR)) contains G(ΔPQR). The result then follows
by Lemma 1.3(2).

These appear to be isolated results. Cabri sketches indicate that it does not generalise to other centres.
We do, however, have some results about the entire circumconics C(ΔPQR).

Corollary 1.3.3
(1) if C(ΔPQR) is the Kiepert Hyperbola of ΔABC, then f(C(ΔPQR)) is the Kiepert Hyperbola of ΔPQR.
(2) if C(ΔPQR) is the Circumcircle of ΔABC, then f(C(ΔPQR)) is the Cirumcircle of ΔPQR.
(3) if C(ΔPQR) is the Steiner Ellipse of ΔABC, then f(C(ΔPQR)) is the Steiner Ellipse of ΔPQR.
Proof
(1) The Kiepert Hyperbola is the unique circumconic containing G and H. The result follows as in Corollaries
1.3.1 and 1.3.2.
(2) The infinite points of the Circumcircle of ΔABC also lie on f(ΔPQR), so this is the Circumcircle of ΔPQR.
(3) The affine map f maps centroids to centroids, and conic centre to conic centre.

Observation
Cabri suggests the following :
If F is a Fermat Point of ΔABC and ΔABC is F-logic with ΔPQR, then the F-point is a Fermat point of ΔPQR.
It is not easy to see which Fermat point corresponds to which. Note that Corollary 1.3.3(1) applies when
both Fermat Points are on C(ΔPQR) - so that this is the Kiepert Hyperbola. It follows that the F-points must
lie on the Kiepert Hyperbola of ΔPQR.

The following results are included since they involve constructions of various sorts.

Lemma 1.4
Suppose that P, Q, X are fixed points. The points R for which ΔABC is X-logic with ΔPQR lie on a line.
Proof
Let Y be the intersections of the lines through P parallel to AX and through Q parallel to BX.
Then Y must be the X-point. It follows that R must lie on the line through Y parallel to CX.

Lemma 1.5
Suppose that P and Q are fixed points, and that C is a circumconic of ΔABC.
There is a unique point R for which C = C(ΔPQR).
Proof
A circumconic is identified by any two of its points other than A,B,C. We choose points X1, X2 on C.
We can apply Lemma 1.4 to X1 and X2 in turn. The required point R is the intersection of the lines
given by the Lemma. Note that, as X1 and X2 lie on a circumconic, the lines CX1, CX2 are non-parallel.
This guarantees that the lines are not coincident, so that the point R is unique.

We noted in Lemma 1.3 that we have two conics with the same infinite points. This is a useful idea.

Definition Conics C1 and C2 are asymptotic if the meet the line at infinity in the same two points.

When the points coincide, we have a parabola with this as centre.

Theorem 1.6
Suppose that P is a fixed point, that C is a circumconic of ΔABC, and that C1 is conic asymptotic with C.
As Q varies along C1, the points R such that C = C(ΔPQR) lie on another conic asymptotic with C.

This is proved by Maple. The proof depends on results in Appendix 2.

2. A result on antipodal points.

The idea of conjugate directions comes from affine geometry. We need some elementary results to prove a
new result on X-logic triangles. These appear in Appendix 3.

Theorem 2.1
Suppose that X1, X2 are antipodal points on the central conic C(ΔPQR), then the X-points are antipodal points
on f(C(ΔPQR)).
Proof
As X1,X2 are antipodal, Appendix 3 shows that each of the pairs AX1,AX2, BX1,BX2 and CX1,CX2 specify
conjugate directions, both for C(ΔPQR) and for f(C(ΔPQR)).
As X1 is on C(ΔPQR), there is point Y1 - the X-point of X1- on f(C(ΔPQR)) such that the lines PY1, QY1, RY1
are parallel to PX1, QX1, RX1, respectively. It follows that the lines through P,Q,R parallel to PX2, QX2, RX2
meet at the antipode of Y1 on f(C(ΔPQR)). This is then the X-point of X2.

3. Triply X-logic triangles.

Theorem 3.1
If ΔABC is X-logic with ΔPQR and with ΔQRP then it is X-logic with ΔRQP.

When the condition of the Theorem holds, we say that ΔABC is triply X-logic with ΔPQR.

The following result is well-known for triply orthologic triangles, but is actually true in general.

Theorem 3.2
If ΔABC is triply X-logic with ΔPQR, then the points of concurrence lie on the Steiner Ellipse of ΔPQR.

4. Notes on triply perspective triangles.

Theorem 4.1
If ΔABC is perspective with ΔPQR and with ΔQRP then it is perspective with ΔRQP.

In such a case, we say that the triangles are triply perspective. A case of special interest is when
ΔPQR is inscribed in a circumconic of ΔABC. We sum up some well-known results in this case. We
begin with a generalisation of the idea of the circumcevian triangle.

Definition
Given a circumconic C and a point X, the C-circumcevian triangle of X is the triangle whose vertices
are the second intersections of C with AX, BX, CX.

Theorem 4.2
Let C be the circumconic with perspector P. Let ΔA'B'C' be the C-circumcevian triange of X.
Then ΔABC is triply perspective with ΔA'B'C' if and only if X is on the tripolar of P.
When X lies on the tripolar of P, we also have the following.
(1) The perspectors of ΔABC with ΔB'C'A' and ΔC'A'B' also lie on the tripolar of P.
(2) The inconic I of ΔABC with perspector P also touches the sidelines of ΔA'B'C'.
(3) With respect to ΔA'B'C', C and I also have perspector P.
(4) The polar of P in C is the tripolar of P.
(5) If P = p:q:r and A' is u:v:w, then B' = pv/q:qw/r:ru/p and C' = pw/r:qu/p:rv/q.
(6) The centriods of the family of ΔA'B'C' lie on the conic
3(pq+qr+rp)(pyz+qzx+rxy)-(p+q+r)(x+y+z)(qrx+rpy+pqz).

Note that, when P is at infinity, this is C itself, see 5.1 below.
When P is on the Steiner Ellipse, it is the union of the line at infinity and the tripolar of P.
Of course, an infinite point cannot occur as a centroid (unless all vertices are at infinity).

5. X-bilogic triangles

Definition
ΔABC is triply X-bilogic with ΔPQR if ΔABC is perspective and X-logic with each of ΔPQR, ΔQRP, ΔRPQ.

After Ehrmann's result on H-bilogic triangles inscribed in the circumcircle, we look for X-bilogic triangles
inscribed in a circumconic. There are several cases of interest.

After Theorem 4.2, we can write down the coordinates of the vertices of the general triangle ΔA'B'C'
inscribed in the circumconic C and triply perspective with ΔABC.

As in section 1, provided that the triangles are not homothetic, the set of points X such that ΔABC is
X-logic with ΔA'B'C' is a circumconic C1 of ΔABC. Likewise the set of X for which ΔABC is X-logic to
ΔB'C'A' is a circumconic C2. In general, there will be just one point X (other than A,B,C) for which
these pairs are X-logic, and hence by Theorem 3.1 the triangles are triply X-logic. The exceptions
arise precisely when one of C1,C2 is the whole plane (so the triangles are homothetic) or the conics
coincide.

We get a unique solution for X unless the coordinates simplify to 0:0:0. Using resultants, this will occur
only if the perspector of C is on :
(a) the line at infinity,
(b) the Steiner Ellipse, or
(c) a conic whose only real point is G = 1:1:1.

Theorem 5.1
Suppose that ΔABC and ΔA'B'C' are inscribed in a circumconic C with perspector P = p:q:r.
If ΔABC and ΔA'B'C' are triply X-logic, then X lies on the quadric
Q : pyz(x2-yz)+qzx(y2-zx)+rxy(z2-xy).

Note
The quadric Q is the isotomic conjugate of the conic Q' : p(x2-yz)+q(y2-zx)+r(z2-xy).
This in turn is the Steiner Inverse of the line px+qy+rz = 0. Thus, we see that

Q is the isotomic conjugate of the Steiner Inverse of the isotomic conjugate of C.

the centroid G,
the P-"orthocentre" - the isotomic of the anticomplement of P,
the P-"brocard points" - the points rp:pq:qr and pq:qr:rp,
the infinite points of the Steiner Ellipse and those of C.

The conic Q' has centre p-2q-2r:q-2r-2p:r-2p-2q - the analogue of X(599).
This is the mid-point of G and the anticomplement of P.
Q' contains G and the anticomplement of P. It also contains q:r:p and r:p:q.
Since it contains the reflections of these in the centre, the conic can be drawn.
It is asymptotic with The Steiner Ellipse.

Case 5.1 Circumconics with perspector P at infinity.
In this case, the quadric Q degenerates into the Steiner Ellipse and the conic C.
As shown in Appendix 1(1), there is a special triangle ΔA'B'C' inscribed in C which is homothetic and
triply perspective with ΔABC. For this triangle, ΔABC is triply X-logic with ΔA'B'C' for every X on the
Steiner Ellipse of ΔABC. The points of concurrence are the reflection of X in the centre of C, and the
second intersections of the Steiner Ellipse of ΔA'B'C' with the lines from X to the common points of
the Steiner Ellipses (and tripolar of P).
For each other triangle ΔA'B'C' on C triply perspective with ΔABC, the triangles are triply X-logic
where X is the centroid of ΔA'B'C'. This point lies on C. We also note that the Steiner Ellipses of all
the ΔA'B'C' all pass through the intersections of The Steiner Ellipse of ΔABC and the tripolar of P.

Case 5.2 Circumconics with perspector P on the Steiner Ellipse.
Once again we have a special triangle ΔA'B'C' as in Appendix 1(2). In this case, the "triangle" has
two vertices at infinity, so for all points X, ΔABC and ΔA'B'C' are trivially triply X-logic.
Suppose now that ΔA'B'C' is inscribed in C and triply perspective with ΔABC. Then the Steiner
Ellipse of ΔA'B'C' passes through P and the fourth intersection of C with the Steiner Ellipse of ΔABC.
In each case, the triangles ΔABC and ΔA'B'C' are triply G-bilogic.
The centroids of triangles ΔA'B'C' lie on the tripolar of P.

Case 5.3 The Steiner Ellipse as circumconic C.
Now all triply perspective triangles inscribed in C are triply G-bilogic with ΔABC.
Note that G is the only real point on the quadric Q.

Case 5.4 The general case.
>From Theorem 5.1, we know that we can have triply X-logic triangles only if X is on the quadric Q.
>From our description of Q, we can parametrize it as follows. Suppose that C has perspector P = p:q:r.
Let U = u:v:w be a point on px+qy+rz = 0. Then u is linear in v,w. Then U*, the isotomic conjugate of
the Steiner Inverse of U is a typical point of Q. Now we write down the algebraic conditions to ensure
that ΔABC is U*-logic with ΔA'B'C' and ΔB'C'A'. These turn out to have the form KL=0, KM=0, with
K linear in v,w and cubic in the coordinates of one of the perspectors, and L,M linear in v,w and the
coordinates. Now, L and M are consistent only when P is at infinity - see case 5.1. The possibility that
K = 0 is trivial arises only if P is on the Steiner Ellipse (case 5.2) or P = G (case 5.3).

Theorem 5.2
Suppose C is the circumconic with perspector P, other than those in cases 5.1, 5.2, 5.3.
For each X on Q, there is a unique triangle inscribed in C triply X-bilogic with ΔABC.

Case 5.5 The case X = G.
Looking at the conditions as above, we find that we can have triply G-bilogic triangles only when
the perspector P is G or on the Steiner Ellipse. See cases 5.2 and 5.3.

Now an important result. This was observed by Cabri. It has been verified by Maple.

Theorem 5.3
Suppose that X is not the centroid G.
Let X* denote the isotomic conjugate of the Steiner Inverse of the isotomic conjugate of X.
There is a triangle triply X-bilogic with ΔABC inscribed in a circumconic C if and only if X*
lies on C.
When X* is on C, the triangle is unique. It is the only inscribed triangle triply perspective
with ΔABC whose Steiner Ellipse has X* as fourth intersection with C.

When X = H, X* is X(98), the Tarry Point. Two of the possible circumconics are the Circumcircle
and the Kiepert Hyperbola. The first has been studied by Ehrmann. The latter is an example of
case 5.1. There, we noted that we get a triply X-logic triangle when X is the centroid of the triangle.
Thus the triply bilogic triangle occurs when this centroid is the orthocentre of ΔABC.
In this case, we equally have that the orthocentre of the inscribed triangle is the centroid of ΔABC.
In general, the condition is equivalent to the X* of the inscribed triangle being on the Steiner Ellipse
of ΔABC. This can be constructed as below.

Case 5.1 revisited.
Suppose that X is a finite point other than G.
Here, we show how to construct a triply X-bilogic triangle inscribed in a certain circumconic.
Let C be the circumconic through X and G. Then the perspector P is at infinity. We have Case 5.1.
It is easy to verify that the associated point X* also lies on C.
By Theorem 5.3, there is a unique triangle ΔA'B'C' inscribed in C triply X-bilogic with ΔABC.
>From the theorem, X* lies on the Steiner Ellipse of ΔA'B'C'. From 5.1, this ellipse also passes
through the intersections Y,Z of the Steiner Ellipse of ΔABC with the tripolar of P.
Also from 5.1, the required ΔA'B'C' has centroid X.
The Steiner Ellipse of ΔA'B'C' must pass through the reflections of X*,Y,Z in X.
This allows us to construct the ellipse S. The points A',B',C' are the intersections of C, S other
than X*.

Theorem 5.4
Suppose that C is a circumconic for ΔABC other than the Steiner Ellipse.
Suppose that ΔPQR is inscribed in C and triply perspective with ΔABC. See Note 1.
Let S be the fourth intersection of C with the Steiner Ellipse of ΔPQR.
Let X be the isotomic conjugate of the Steiner Inverse of the isotomic conjugate of S.
Then ΔABC is triply X-logic with ΔPQR.

Note 1.
We have problems when C passes through G - so has perspector at infinity - and ΔPQR is the
reflection of ΔABC in the centre of C. Then the triangles are homothetic so are X-logic for all X.
In this case, The Steiner Ellipse of ΔPQR meets C at S = G. The related X is undefined.

This is quickly verified by Maple, but also follows from Theorem 5.3.

Appendix 1 - some special circumconics.

(1) Let P = p:q:r be a point at infinity, and let C be the circumconic with perspector P.
Let A' = qr:q2:r2, B' =p2:pr:r2, C' = p2:q2:pq.
Then ΔABC and ΔA'B'C' are triply perspective with perspectors p2:q2:r2, rp:pq:qr and pq:qr:rp.
Note that the second and third perspectors are the intersections of both Steiner Ellipses with the
tripolar of P.
The first perspector is the centre of C, so that ΔA'B'C' is ΔABC rotated by π about this point.

(2) Let P = p:q:r be a point on the Steiner Ellipse, and let C be the circumconic with perspector P.
Let A' = p2:q2:r2, B' = rp:pq:qr, C' = pq:qr:rp.
Then ΔABC and ΔA'B'C' are triply perspective with perspectors qr:q2:r2, p2:pr:r2, p2:q2:pq.
Note that B' and C' are the infinite points on C.
The point A' is on C,but also on the line joining P to the fourth meet of C with the Steiner Ellipse
of ΔABC. This identifies A'.

Appendix 2 - conjugate directions.

The concept of conjugate directions arises naturally in affine geometry. A direction in the plane can
be thought of as a equivalence class under the relation parallel.

Theorem A2.1
Suppose that C is a central conic - an ellipse or hyperbola. Let X be a direction not parallel to an asymptote
of C in the case of a hyperbola.
Given the direction X, the mid-points of the chords of C in this direction lie on a diameter of C.
Further, if this diameter has direction Y, then the mid-points of chords parallel to Y lie on the diameter
with direction X.

Of course, an affine proof must deal separately with the types of conic.

We say that X and Y are conjugate directions.

Theorem A2.2
Suppose that A,B,C are points on a central conic C.
The lines AB and AC define conjugate directions if and only if BC is a diameter.

In fact, neater proofs may be obtained from projective geometry.

Suppose that C is a conic and L is a line not tangent to C. We say that P,Q on L are L-conjugate when
the polar of P in C contains Q. Equally, we may take the condition as the polar of Q contains P.
The intersections of L and C are self-conjugate as the polar of a point on C is the tangent.
For a line M other than L, we say that the L-direction of M is the intersection of L and M.

Here, since the concepts of pole and polar are clearly projective, we can assume that C is yz+zx+xy = 0,
and indeed any three points on C can be taken as X[1,0,0], Y[0,1,0] and Z[0,0,1].

Lemma A2.3
If X and Y lie on a conic C, and P ≠ X,Y is on XY. Let Q be the intersection of XY with the polar of P.
Then (X,Y,P,Q) = -1.
Proof
With the above assumptions, P has the form [a,b,0]. Its polar is bx+ay+(a+b)z = 0, and Q is [-a,b,0].
The result is immediate.

Lemma A2.4
Suppose that X,Y,Z are points on a conic C, with X not on L.
The L-directions of XY and XZ are L-conjugate if and only if YZ contains the pole of L in C.
Proof
With the above assumptions, as X is not on L, L has equation ax+by+cz = 0, with a ≠ 0.
Now, XY is z = 0, so the L-direction of XY is [-b,a,0], with polar ax-by+(a-b)z = 0.
As XZ is y = 0, the L-direction is [-c,0,a]. This is on the polar if and only if a(a-b-c) = 0.
The pole of L is [-a+b+c,a-b+c,a+b-c]. As a ≠ 0, the result is equivalent to the pole being on YZ.

We now return to the affine theory, but now think of the projective plane as the affine plane plus L*,
the line at infinity. We note that the L*-direction of an affine line M is its "point at infinity". Two affine
lines are parallel means that they have the same L*-directions. Further, if (X,Y,P,Q) = -1 and P is on
the line L*, then Q is the mid-point of segment XY. The centre of a conic is simply the pole of L*.
We can now prove Theorems A2.1 and A2.2.

Proof of Theorem A.2.1
By Lemma A2.3 and the above remarks, the polar of X bisects all chords with direction X.
As X is on L*, this polar passes through the pole of L* - the centre of C. This polar is the required
diameter. The diameter has direction Y. As this is on the polar of X, La Hire shows that X is on the
polar of Y. This proves tha last sentence of the Theorem.

Note that this also proves that the affine notion of conjugacy is equivalent to L*-conjugacy.

Proof of Theorem A2.2
Since we are really dealing with L*-conjugacy, Lemma A2.4 gives the result provided that the point
A is not on L*. As the Theorem is affine, this condition is automatically satisfied.

Appendix 3 - asymptotic conics.

Note that this is the notion of "shaped conics" described by Steve Sigur. To my mind, the term
"shaped" suggests conics of the same eccentricity. In fact, such conics need not have the same
eccentricity - we can have two hyperbolas with the same asymptotes, but lying on opposite sides
of the asymptotes. Unless they are rectangular, the eccentricities are different.

Consider the class of conics through two fixed points at infinity. As a conic is specified by five points,
the class contains a single circumconic.

Theorem A3.1
The members of a class of asymptotic conics have equations k(pyz+qzx+rxy)+(x+y+z)(ux+vy+wz) = 0,
where pyz+qzx+rxy = 0 is the circumconic of the family.

It is clear that asymptotic conics have parallel axes. It is also clear that homothetic conics are asymptotic.
The converse is not true as the above example of two hyperbolas shows. However, if we work in complex
coordinates, then we can show that asymptotic non-degenerate conics are related by a "homothety" of
the form H(X,k), with k2 real and non-zero.

Here, we are interested in the concept of conjugacy for asymptotic conics.

Theorem A3.2
Asymptotic non-parabolic conics have the same pairs of conjugate directions.
Proof
A non-parabolic conic is one with distinct points at infinity.
Here it is convenient to use the matrix form for equations of conics and polars. A conic has an equation
of the form f(x,x) = xTMx = 0. The polar of the point A = a has equation f(a,x) :aTMx = 0.
Now suppose that have asymptotic conics with distinct infinite points A = a and B = b.
Note that this means that f(a,a) = f(b,b) = 0.
Any infinite point U other than A and B has the form ua+vb, with uv ≠ 0.
Let U' be the general infinite point u'a+v'b.
Then U, U' denote conjugate directions if and only if f(u'a+v'b,ua+vb) = 0.
By elementary manipulation, this is equivalent to (uv'+u'v)f(a,b) = 0.
We now show that f(a,b) ≠ 0. If it is zero, then B is on the polar of A. But this is the tangent at A.
But this means that the tangent at A is the line at infinity, contrary to the hypothesis.
Thus U,U' conjugate if and only if uv'+u'v = 0. As uv ≠ 0, (u',v') must be a multiple of (u,-v).
We have therefore shown that the conjugate pairs are determined solely by the infinite points.