Pedal and Antipedal Points

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.