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}.

Definitions
A point P is an {X}-pedal point if P belongs to the class {X} with respect to its pedal triangle.
If P belongs to the class {Z} with respect to the reference triangle, the class {Z} is the pedal class of {X}.
A point P is an {Y}-antipedal point if P belongs to the class {X} with respect to its antipedal triangle.
If P belongs to the class {Z} with respect to the reference triangle, the class {Z} is the antipedal class of {X}.
For a point Q, g(Q) denotes the isogonal conjugate of Q in the reference triangle.

Of course, if {Z} is the pedal class of {W}, {W} is the antipedal class of {Z}. We now concentrate on pedal
triangles since the corresponding results for antipedal triangles can now be easily deduced.

Our first result shows that the theory for pedal triangles can be deduced from that for circumcevian triangles.
The webpage Circumcevian Triangles gives results on the latter.

Theorem 1
Let U be a point not on the Circumcircle or the Line at Infinity.
Let p(Δ), cm(Δ) denote respectively the pedal and circumcevian triangles of U.
(a) The corresponding sides of p(Δ) and cm(Δ) are in the same ratio.
(b) If U has barycentric coorcinates u:v:w with respect to the reference triangle, then its
coordinates with respect to p(Δ) are those of g(U) with respect to the reference triangle.
(c) The barycentric coordinates of U with respect to its p(Δ) are equal to the barycentric
coordinates of the isogonal conjugate of U [in cm(Δ)] taken with respect to cm(Δ)
Proof
These can be proved by direct calculation.

Corollary 1.1
(a) {Z} is the pedal class of {X} if and only if {Z} is the circumcevian class of {g(X)},
(b) {Z} is the pedal class of {X} if and only if {g(X) is the pedal class of {g(Z)}.

Part (b) follows from (a) and the fact that if {W} is the circumcevian class of {Y},
then {Y} is the circumcevian class of {W}.

We shall use this later to find results about curves.

extraversion class	pedal class	notes
{X1}	{X4}
{X2}	{X6}
{X3}	{X1}
{X4}	{X3}
{X6}	{F1,F2}	F1, F2 are the real foci of the Steiner Inellipse
{X15,X16}	{X13,X14}
{X80}	{X186}	{X80} has four members, X186 is strong
{X265}	{X36}	{X36} has four members, X265 is strong
{X671}	{X187}	X187, X671 are strong
{B1}	{B1}	B1, B2 are the Brocard Points
{B2}	{B2}}	B1, B2 are the Brocard Points
{F1,F2}	{X2}	F1, F2 are the real foci of the Steiner Inellipse

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).

Definitions
For a strong curve F,
p(F) = { P : P is a pedal point for some Q on F }.
ap(F) = { P : P is an antipedal point for some Q on F }.
cm(F) = { P : P is a circumcevian point for some Q on F }.

Note that, for example, p(F) is equally the locus { P : P is on the curve F with respect to its pedal triangle }.
After Theorem 1(a), we note that the equation of F relative to p(Δ) is the same as that relative to cm(Δ).

As before, we can identify p(F) as follows. Let ΔA'B'C' be the pedal 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 pedal 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.

>From Theorem 1, we can relate p(F), ap(F) and cm(F) as follows.

Theorem 2
Suppose that F is a strong curve.
(a) p(F) = cm(g(F)),
(b) ap(F) = g(cm(F)).

We begin with some pairs where the curves F and p(F) differ, but both are well-known.
The first two of these appear in the relevant sections of CTC - related curves

curve F	description	curve p(F)	description
LE	Euler Line	Q002	Euler-Morley Quartic (circular)
LB	Brocard Axis	Q039	bicircular isogonal sextic
JH	Jerabek Hyperbola	Q037	inversible bicircular quintic
KH	Kiepert Hyperbola	LB	Brocard Axis
SE	Steiner Ellipse	LL	Lemoine Axis
Q039	bicircular isogonal sextic	KH	Kiepert Hyperbola
Q003	Euler-Morley Quintic	JH	Jerabek Hyperbola
Q030	a bicircular septic	LE	Euler Line
K007	Lucas Cubic	K172	pK(X32,X3) = g(K007)
K015	Tucker Nodal Cubic	K229	cK(#X6,X6) = g(K015)
K016	Tucker + Cubic		nK0(X32,X6) = g(K016)
K060	Kn	K073	Ki = g(Kn)
K064	Sharygin Cubic	K148	nK0(X50,x6) = g(K064)
K181	pK(X4,X4)	K167	pK(X184,X6) = g(K181)
AC-X	X-Apollonian Circle, X = A,B,C	K053-X	X-Apollonian Strophoid = g(AC-X)

Definition
A strong curve F is a (anti)pedal invariant curve if p(F) = F (then ap(F) = F also).

Some Pedal/Antipedal Invariants

name	description	notes
K001	Neuberg Cubic	Note 1
K002	Thomson Cubic	Note 1
K003	McCay Cubic	Note 1
K004	Darboux Cubic	Note 1
K005	Napoleon Cubic	Note 1
K006	OrthoCubic	Note 1
K018	Brocard Second Cubic	Note 2
K024	Kjp Cubic	Note 2
K156	Soddy-Euler Cubic	Note 1
Q021	bicircular isogonal sextic	Note 3

Note 1
These cubics lie in the Euler Pencil. Indeed an isogonal pedal invariant pK must lie in this pencil. The argument
is precisely as in Circumcevian Triangles.

Note 2
These cubics generate a pencil of isogonal pedal invariant nK0.

Note 3
The sextic is isogonal and circumcevian invariant, and hence pedal invariant.

A curiousity
>From the first table above, we note that X2 is a focus of the inellipse of its pedal triangle T2. The second focus
is then the isogonal of X2 in T2. Using standard results on change of coordinates, we find that the barycentrics
are f(a,b,c);f(b,c,a):f(c,a,b), with f(a,b,c) = a²(b⁴+c⁴-3b²c²-a²b-a²c). The mid-point of the foci is the centre
of the inellipse, and hence the centroid of T2. It is easily computed, and is X373 in ETC. Thus the second focus
is the relection of X2 in X373, and lies on X2X373 = X2X51.

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