As in Table 28, we consider points defined over an algebraic extension of the field generated by a^{2}, b^{2}, c^{2}.

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(a^{2},b^{2},c^{2},x,y,z) = F(Pa,Pb,Pc,PA,PB,PC) = 0, where F has equation F(a^{2},b^{2},c^{2},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) = a^{2}(b^{4}+c^{4}-3b^{2}c^{2}-a^{2}b-a^{2}c). 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.