Circumcevian Triangles

As in Table 28, we consider points defined over an algebraic extension of the field generated by a2, b2, c2.
As before, the set of algebraic extraversions (i.e. algebraic conjugates) of such a point X is denoted by {X}.

A point P is an {X}-circumcevian point if P belongs to the class {X} with respect to its circumcevian triangle.
For such a P, we refer to the class {P} as the circumcevian class of {X}.

In Hyacinthos #11942, Chia-Lin Hwang introduced the notation Xcm(P|Q) = R to express the fact that the
point P with respect to ΔABC is the point R with respect to the circumcevian triangle of Q. As in Table 28, it
is usually only possible to say that P belongs to a class {R} with respect to the circumcevian triangle of Q.
For example Xcm(X4|X4) is always in {X1}, but is X1 only if ΔABC is acute angled.

With the definition above, P is an {X}-circumcevian point if and only if Xcm(P|P) is in {X}.
Thus, X4 is the only {X1}-circumcevian point. We now observe that Xcm(P|P)=R if and only if Xcm(R|R) = P.
Thus, the members of {X1} are the X4-circumcevian points.

Most of the following facts appear in various Hyacinthos mesages, although the accounts of X1, X13 do not
indicate that any of the circumcevian class may occur depending on the shape of the reference triangle.
The results on X36, X186 arise by inversion of X1, X4 in the Circumcircle, and the following result. Some of
this is well-known, the rest is new to me.

For a finite point P other than X3, Pi denotes the inverse of P in the Circumcircle.

Theorem 1
Let U be a finite point other than X3.
(a) The circumcevian triangles of U and Ui are reflections in the line UX3.
(b) The point U has the same baycentric coordinates for these circumcevian triangles.
(c) If U is a {V}-circumcevian point, then Ui is a {Vi}-circumcevian point.
Direct calculation establishes (b) and the fact that the lengths of the corresponding sides
of the circumcevian triangles are equal. The latter is all we need from (a), but (a) is true.
Now suppose that Xcm(U|U) = V. By (b), Xcm(U|Ui) = V. We observe that the reference
triangle and the circumcevian triangles have the same Circumcircle. The geometry now
shows that Xcm(Ui|Ui) = Vi, which establishes (c).

We shall use this later to find results about curves, see Corollary 1.1.

 extraversion class   cicumcevian class  notes    
{X1 = I, Ia,Ib,Ic} {X4}  
{X2} {F1,F2} F1, F2 are the real foci of the Steiner Inellipse
{X3} {X3}  
{X4} {X1=I,Ia,Ib,Ic}  
{X6} {X6}  
{X13,X14} {X13,X14}  
{X36} {X186} X186 is strong
{X186} {X36} {X36} has four members
{X187} {X187}  
{B1} {B2} B1, B2 are the Brocard Points
{B2} {B1} B1, B2 are the Brocard Points
{F1,F2} {X2} F1, F2 are the real foci of the Steiner Inellipse
{Q(φ),Q(-φ)} {Q(φ),Q(-φ)} Q(φ) = sin(A)sin(A+φ), φ real

From Theorem 1, the circumcevians of X23 are the inverses of the Steiner foci.

The isodynamic points {X15,X16} are the points for which the circumcevian triangles are equilateral.
It follows that these points have no circumcevian points.

A strong point P is a circumcevian invariant point if Xcm(P|P) = P.

Trivially, X3 is circumcevian invariant.
From Hyacinthos #11942, X6 and X187 are circumcevian invariant points.
The geometry shows that any real centre X at infinity is a circumcevian invariant point. Then the circumcevian
triangle of X is the reflection of the reference triangle in the line LX through X3 perpendicular to the direction X.
The reflection of a centre Y in LX is the centre Y for the reflected triangle. The "reflection" of X is X.

In Table 28, we noted that, if F is a strong curve, then the locus of cevian (resp. anticevian) points of points
on F lie on a strong curve c(F) (resp. a(F)). We found curves with F = c(F) = a(F). We consider the circumcevian
points for points on F. These lie on the strong curve cm(F), which we can identify as in Table 28. Observe that
we have cm(cmF) = F. Once again, we look for curves with cm(F) = F.

As before, we identify cm(F) as follows. Let ΔA'B'C' be the circumcevian triangle of a point P. Let Pa, Pb, Pc
be the squares of the sidelengths of ΔA'B'C', and let PA:PB:PC be barycentric coordinates for P with respect
to ΔA'B'C'. Then the condition that P is a circumcevian point for a point on the strong curve F is given by the
equation PF(a2,b2,c2,x,y,z) = F(Pa,Pb,Pc,PA,PB,PC) = 0, where F has equation F(a2,b2,c2,x,y,z) = 0. We may
remove factors which correspond to degenerate circumcevian triangles to get an equation for cm(F).

We begin with three pairs where the curves F and cm(F) differ, but both are well-known.

curve F description curve cm(F) description
LE Euler Line Q037 inversible bicircular quintic
JH Jerabek Hyperbola Q002 Euler-Morley Quartic (circular)
KH Kiepert Hyperbola Q039 bicircular isogonal sextic

A strong curve F is a circumcevian invariant curve if cm(F) = F.

Some Notation
In practice, we find that, for such a curve F(a2,b2,c2,x,y,z) = 0 the equation PF(a2,b2,c2,x,y,z) = 0 computed
above has the form K(C)kF(a2,b2,c2,x,y,z) = 0, where C is a fixed expression symmetric in a2,b2,c2 and in x,y,z,
K is a real number, the constant of the curve, as we see below, K = 1 except in one case.
k is a positive integer, the power of the curve
The curves we consider have equations polynomial in a2,b2,c2,x,y,z
homogeneous in a2,b2,c2, the degree being the a2-degree of the curve and also
homogeneous in x,y,z, the degree being the x-degree of the curve.

We write Fi for the inverse of a curve F in the Circumcircle. From Theorem 1, we now have the

Corollary 1.1
For any strong curve F, cm(Fi) = cm(F)i.
If F is circumcevian invariant, then so is Fi.

As an example, the Lemoine Axis has inverse the Brocard Circle, so the fact that the former is circumcevian
invariant shows that the latter is also circumcevian invariant.
We also observe that the Euler Line is inversive, so it follows that Q037 is also inversive. As any real infinite
point is invariant, we can deduce the directions of the real asymptotes of Q002, Q037, Q039. The other infinite
points must be circular.

Some Circumcevian Invariants

name description x-degree a2-degree constant power notes
Σ Σ a4-b2c2 0 2 1 2  
Π a2b2c2 0 3 1 2  
LA Sideline BC : b2c2x = 0 1 2 1 2 note the equation
LB Sideline CA : c2a2y = 0 1 2 1 2 note the equation
LC Sideline AB : a2b2z = 0 1 2 1 2 note the equation
KA Symmedian AK : a2(c2y-b2z) = 0 1 2 1 2 note the equation
KB Symmedian BK : b2(a2z-c2x) = 0 1 2 1 2 note the equation
KC Symmedian CK : c2(b2x-a2y) = 0 1 2 1 2 note the equation
LI Line at Infinity 1 0 -1 1  
LL Lemoine Axis 1 2 1 2  
LBr Brocard Axis 1 3 1 3  
CC Circumcircle 2 1 1 2  
BrC Brocard Circle 2 2 1 3  
BrI Brocard Inellipse 2 3 1 4  
Δ union of sidelines : xyz = 0 3 0 1 2  
pK(X577,X3) union of X3-cevians 3 6 1 7  
pK(X32,X6) union of X6-cevians 3 3 1 4  
K001 Neuberg Cubic 3 3 1 5 Note 1
K002 Thomson Cubic 3 1 1 3 Note 1
K003 McCay Cubic 3 3 1 5 Notes 1,4
K004 Darboux Cubic 3 3 1 5 Note 1
K005 Napoleon Cubic 3 3 1 5 Note 1
K006 OrthoCubic 3 3 1 5 Note 1
K018 Brocard Second Cubic 3 2 1 4 Note 2
K024 Kjp Cubic 3 2 1 4 Notes 2,8
K053-A,-B,-C Apollonian Strophoids 3 2 1 4 Note 3
K073 Ki = pK(X50,X3) 3 6 1 7 Notes 4,7
K114 iK(X74) 3 6 1 7 inversible
K148 nK0(X50,X6) 3 5 1 6 Note 8
K156 Soddy-Euler Cubic 3 3 1 5 Note 1
K167 pK(X184,X6) 3 4 1 5  
K172 pK(X32,X3) 3 4 1 5 Note 4
K229 cK(#X6,X6) 3 3 1 4 Note 5
K349 pK(X2965,X3) 3 6 1 7 Note 4
Q019 circular quartic 4 2 1 7 Note 6
Q021 bicircular isogonal sextic 6 5 1 9 Note 6
Q024 bicircular sextic 6 5 1 9 Note 6
Q047 McCay Quintic 5 6 1 9 Note 7
Q049 Apollonian Quartic 4 5 1 7 Note 7

Proposition 1
Suppose that a set {Fi} of circumcevian invariant curves have equations of the form Fia2,b2,c2,x,y,z) = 0,
each with the same values of x-degree, a2-degree, constant K, and power k. Then, for real constants ni,
the curve ΣniFi is also circumcevian invariant, and has the same values of these parameters.

The proof is obvious.

Note that we require real multipliers. For example, with two such curves F, G, we get some of the members
of the pencil generated by F and G.

Using the information from the table above, we can obtain many families of circumcevian invariant curves.
We comment only on examples where the family is in the literature, or has particularly simple geometry.

A triangle centre P on the Euler Line is an [O,H]-point if OP:PH = m:n, with real m,n.

Note 1 - a "pencil" of isogonal pK Euler Pencil. See also the euler-brocard-jerabek net
From the table K001,K002,K003,K004,K005,K006,K156 and the cubic with equation obtained by multiplying
the usual equation for K002 by the constant Σ all have the same values of the parameters. In fact, they all
belong to the Euler Pencil discussed in Table 27 of CTC.
The circumcevian invariant curves described in Proposition 1 are precisely the cubics of the form nK(X6,P),
where P is an [O,H]-point.

Theorem 2
An circumcevian invariant isogonal pK belongs to the Euler Pencil.
Suppose that F is a circumcevian invariant isogonal pK.
As F is a pK it contains each member of {X1}.
Then, since F is circumcevian invariant, X4 is on F.
It follows that the pivot lies on the Euler Line, so F is on the Euler Pencil.

Note 2 - a "pencil" of isogonal nK0
The cubics K018 and K024 have the same values of the parameters, so that we can again use Proposition 1.
They are also isogonal cubics of type nK0. Thus we obtain some isogonal nK0 with roots on X6X523.
The cubic K018 is the circular member.

Note 3 - a "net" of cubics through the Fermat Points
To get the equations for the strophoids in the form in the table, we must multiply the usual equation for
the A-Strophoid by a2, and similarly for the B- and C-Strophoids. We can then apply Proposition 1. Now we
obtain certain cubics in the net generated by the K053. This net is discussed in the K053 page, and includes
K001, K018 and K060. By looking at the values of the parameters, only K018 appears from proposition 1.
The others require coefficients which are merely symmetric functions rather than real numbers. In fact,
K001 is circumcevian invariant, but K060 is not.

Note 4 - a "pencil" of pK(W,X3). See also two brocard pencils and euler-brocard-jerabek net
If we multiply the equation for K003 by (abc)2, it has the same parameters as for K073. Thus we have another
"pencil". The poles are the points on the Brocard Axis, as described in Note 8. We also have W = X32, giving the
cubic K172 (equation multiplied by the constant Σ of the table), W = X2965, giving K349, W = X571 and W= X577.
The last gives the degenerate cubic consisting of the X3-cevians.
This is part of Gibert's Class CL021

Note 5 - a "pencil" of nK with pole X32, root X6
If we take the sidelines as the degenerate cubic with equation a2b2c2xyz = 0, then it has the same parameters
as K229 = nK(X32,X6,X6). Applying Proposition 1, we get certain cubics of type nK(X32,X6,?). This will include
the cubic nK0(X32,X6), the isogonal conjugate of K016.
In fact, the pencil is part of a larger family. If we define the degenerate cubic XYY:=a2c4xy2 = 0, and XZZ, YXX,
YZZ, ZXX, ZYY similarly. Then each turns out to be a circumcevian invariant curve with the same parameters as
K229. Thus Proposition 1 yields a six-parameter family of cubics. This includes the pK which degenerates as the
union of the symmedians.

Note 6 - a "pencil" of bicircular sextics
The curves Q021, Q024 have the same values of the parameters, so that Proposition 1 can be applied.
We get members of the pencil generated by these curves. These can be described as follows :
Suppose that P is an [O,H]-point (as in Note 1), other than G,O. We can then define the locus
    { X : X, P, X' are collinear, where X' is the centre P of the circumcevian triangle of X }.
This will be a bicircular sextic, as described in the Q024 page. The point H gives Q021, the isogonal
member of the set, the point X5 gives Q024.
The point G gives the circumcevian invariant curve consisting of the union of Q019 and the Circumcircle.
The equation obtained from Proposition 1 is the product of the usual equations for these curves, multiplied
by the constant Σ. This shows that Q019 itself is circular rather than bicircular.
The point O obviously gives the whole plane as locus.

Note 7 - inversion in the circumcircle
From Corollary 1.1, if F is a circumcevian invariant curve, then so is Fi, the inverse of F in the Circumcircle.
The invariant cubics K001, K003, K018 give rise to K073, Q047 and Q049.

Note 8 - a "pencil" of nK0(W,X6) See also two brocard pencils
If we multiply the equation for K024 by (abc)2, it has the same parameters as for K148. Thus we have another
"pencil". Here the poles W are certain points on the Brocard Axis. Apart from X6 and X50, we also have invariant
cubics with W = X32, X577 and X571, X2965. The example nK0(X32,X6) appears also in Note 5.
This is part of Gibert's Class CL022

We define the suitable W in notes 4 and 8 as follows :
Suppose that we take an angle φ in [-π/2,π/2]. Then we have a point P(φ) with first barycentric sin(A)(sin(A+φ).
The points P(φ) and P(-φ) are harmonic with respect to X3, X6. The barycentric product of P(φ) and P(-φ) is then
a point W(φ), also on the Brocard Axis. Indeed, W has first barycentric sin2(A)(k(4cos2((A)-1)-1), where k,φ
are related by k = 1/(2cos(2φ)+1) or cos(2φ) = (1-k)/2k. Then these real φ give our points W.

A curiousity
We have observed that a real focus F of the Steiner Inellipse is the centroid of the circumcevian triangle of F.
This may be regarded as saying that X2 is a focus of the Steiner Inellipse of its circumcevian triangle T2.
The second focus F2 will then be the isogonal of X2 with respect to T2. Using standard results on change of
coordinates, we find the barycentrics of F2 with respect to the reference triangle. This point is not yet in ETC.
The first coordinate is 4a6 -4a2b4 -4a2c4 -2b2c4 -2b4c2 -a2b2c2. The point F2 lies on the line X2X187. This
line is therefore the major axis of the Steiner Inellipse of the circumcevian triangle of X2.