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antigonal

The antigonal N of a point M is equivalently :

  1. the isogonal conjugate of the inverse in the circumcircle of the isogonal conjugate of M,
  2. the antipode of M on the rectangular circumhyperbola passing through M (hence the midpoint of MN lies on the nine-point circle),
  3. the common point of the circles BCA', CAB', ABC' and A'B'C' where A', B', C' are the reflections of M in the sidelines of ABC.
  4. the unique point such that (MA,MB)+(NA,NB)=(MB,MC)+(NB,NC)=(MC,MA)+(NC,NA) = 0 (oriented line angles), definition by Jan van Yzeren, 1941. See also Mathematics Magazine, vol. 65, n°. 5, December 1992.

The antigonal image of a line is, in general, a bicircular quintic. See K186 for details. See also Q030 and CL019 for antigonal cubics.

symgonal

The symgonal Q of a point P is equivalently defined as follows :

  1. if A',B',C' are the reflections of A,B,C through a point P, the circles BCA', CAB', ABC' have a common point Q which is called the symgonal of P. (Jean-Pierre Ehrmann)
  2. the circles AB'C', BC'A', CA'B' have a common point Q' which is also on the circumcircle and the circumconic with center P. Q is the reflection of Q' in P. (Paul Yiu)
  3. Q is the antigonal of the anticomplement of P (or P is the complement of the antigonal of Q).

GST

Consider the mapping f : M(x:y:z) --> M'(x':y':z') given in barycentric coordinates by :

x' =(b^2SB/y - c^2SC/z)x/a^2 ; y' = (c^2SC/z - a^2SA/x)y/b^2 ; z' = (a^2SA/x - b^2SB/y)z/c^2.

When M is a point on the circumcircle, M' is the tripole of the Simson line of M and therefore lies on the Simson cubic K010.

A geometric construction of GST(M) is the following : the trilinear polar of X(69) meets the sidelines of ABC at U, V, W. N is the isotomic conjugate of the isogonal conjugate of M. Its trilinear polar L meets the sidelines of ABC at A', B', C'. The conic (Ca) through A, V, W, A' tangent at A' to BC meets L at A' and M'.

See also the preamble before X(2394) in Clark Kimberling's ETC.

GST transforms any line passing through O into an inscribed conic whose perspector is the isotomic conjugate of the isogonal conjugate of the trilinear pole of the line. Since this tripole lies on the circumconic with center X(6), the perspector is a point of the circumconic with perspector X(69). In particular, GST transforms the Brocard axis into the Kiepert parabola.

GSC

There are always two points M and N such that GST(M) = GST(N) = P. These points are then said to be conjugated under the GSC transformation.

In other words, GST(M) = GST(N) <=> N = GSC(M) <=> M = GSC(N). N is in fact the isogonal conjugate of the cevian quotient of H (orthocenter) and M* (isogonal conjugate of M). See exemples of such pairings in the table below.

Given a fixed point Q, the locus of M such that M, GSC(M), Q are collinear is a pK passing through Q, O = X(3), with pivot P = Ceva point of Q and O (see glossary in ETC), invariant under the isoconjugation which swaps P and O. For example, with Q = X(155), it is K006 (orthocubic), with Q = X(1993), it is K045 (Euler perspector cubic) and with Q = X(6), it is K168.

A simultaneous construction of GSC(M) and GST(M) for M ≠ O is the following. Draw the circumconic (C) passing through M and tangent at M to the line OM. The polar line of O in (C) meets (C) at M and N = GSC(M). The isotomic conjugate of the isogonal conjugate of the perspector of (C) is P = GST(M) = GST(N).

Note that GSC transforms the circumcircle of ABC into the isogonal transform of the nine point circle.

The following table gives a small selection of points M, N and P mentioned above. This excludes points M or N on the circumcircle. When a point is not in the current edition of ETC, the first barycentric coordinate is given as far as it is reasonably simple.

M

N

P

X(1)

X(90)

(b-c)(b+c-a) / a

X(2)

X(69)

(b^2-c^2)SA / a^2

X(4)

X(254)

(b^2-c^2) / (a^2 SA)

X(6)

X(193)*

X(523)

X(32)

 

X(669)

X(48)

a^3 SA / (ab+ac-bc)

X(1459)

X(54)

X(96)

X(1625)*

X(184)

a^4 SA / (a^2b^2+a^2c^2-b^2c^2)

X(3049)

X(485)

X(486)

X(850)

 

 

 

 

 

 

CST

See The Cevian Simson Transformation