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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' = x (b^2 SB / y - c^2 SC / z) / a^2 , y' and z' likewise.

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), also N is the X(3)-crossconjugate of M. See exemples of such pairings in the list 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 Q120, the isogonal transform of the nine point circle. More generally, GSC transforms the circum-conic with perspector X into the isogonal transform of the bicevian conic C(H, X*) where X* is the isogonal conjugate of X.

Peter Moses contributed a selection of these points M, N, P = X(i), X(j), X(k) for these i, j, k with i < j.

i ; j ; k

1 ; 90 ; 4391

2 ; 69 ; 3267

4 ; 254 ; 14618

6 ; 8770 ; 523

21 ; 283 ; 15411

24 ; 14517 ; 15423

25 ; 15369 ; 2489

31 ; 15370 ; 2484

32 ; 15371 ; 669

39 ; 15372 ; 2528

48 ; 15373 ; 1459

54 ; 96 ; 15412

55 ; 15374 ; 4130

56 ; 15375 ; 3669

58 ; 15376 ; 7192

59 ; 100 ; 2397

63 ; 77 ; 15413

71 ; 15377 ; 4064

74 ; 10419 ; 2394

78 ; 271 ; 15416

81 ; 272 ; 15417

95 ; 97 ; 15414

98 ; 2065 ; 2395

99 ; 249 ; 2396

101 ; 15378 ; 2398

102 ; 15379 ; 2399

103 ; 15380 ; 2400

104 ; 15381 ; 2401

105 ; 15382 ; 2402

106 ; 15383 ; 2403

107 ; 15384 ; 2404

108 ; 15385 ; 2405

109 ; 15386 ; 2406

110 ; 250 ; 2407

111 ; 15387 ; 2408

112 ; 15388 ; 2409

184 ; 15389 ; 3049

187 ; 15390 ; 1649

248 ; 15391 ; 879

264 ; 5392 ; 15415

265 ; 15392 ; 14592

284 ; 15393 ; 7253

394 ; 15394 ; 4143

476 ; 15395 ; 2410

477 ; 15396 ; 2411

485 ; 486 ; 850

662 ; 7045 ; 15418

675 ; 15397 ; 2412

895 ; 15398 ; 14977

911 ; 15399 ; 2424

1073 ; 15400 ; 14638

1141 ; 15401 ; 2413

1176 ; 1799 ; 4580

1292 ; 15402 ; 2414

1293 ; 15403 ; 2415

1294 ; 15404 ; 2416

1295 ; 15405 ; 2417

1296 ; 15406 ; 2418

1297 ; 15407 ; 2419

1333 ; 15408 ; 3733

1437 ; 15409 ; 7254

1444 ; 1790 ; 15419

1791 ; 1798 ; 15420

2420 ; 15410 ; 3233

5504 ; 12028 ; 15421

8884 ; 14518 ; 15422

8946 ; 8948 ; 2501

In the red lines, one of the points M, N lies on the circumcircle, the other lies on Q120 and then P lies on K010.

In the blue lines, one of the points M, N lies on the Brocard axis, the other lies on a quintic and then P lies on the Kiepert parabola.

CST

See The Cevian Simson Transformation

Orion transformations OT, OP, OC

See Table 11

Psi and its generalization Psi_P

The Psi quadratic involution is described here. Its is a special (and probably the most interesting) case of the transformation Psi_P defined and studied below. Psi is Psi_X2 or Psi_G. See also the paper Cubics passing through the foci of an inscribed conic.

Let P = p : q : r be a finite point not lying on the sidelines of ABC. Ha is the rectangular hyperbola with center P, passing through A whose asymptotes are the parallels at P to the A-bisectors of ABC. Hb and Hc are defined cyclically.

PsiPhypABC

The equation of Ha is :

(p+q+r) (b^2 z^2–c^2 y^2) + 2 (x+y+z) (c^2 q y–b^2 r z) = 0.

Ha also passes through :

• A' = reflection of A in P,

• Ab = AB /\ reflection of AC in P,

• Ac = AC /\ reflection of AB in P,

• F1, F2 real foci of the inconic with center P and also the imaginary foci F'1, F'2.

The tangent at A to Ha passes through the isogonal conjugate P* of P.

***

Ha, Hb, Hc are obviously in a same pencil since ( a^2 Ha) identically vanishes.

( p x Ha) is the equation of pK(X6, P) which confirms that the three hyperbolas pass through the four foci mentioned above.

More generally, if Q = u : v : w is another point, ( u x Ha) is the equation of spK(Q, P) as in CL055.

***

The Psi_P transform Psi_P(M) of a point M is the pole of M in the pencil above. Psi_P can be seen as the isogonal conjugation in the non proper triangle whose vertices are those of the diagonal triangle of the four foci, namely P and the circular points at infinity J1, J2. See Table 62 for other similar transformations.

It follows that Psi_P is a quadratic involution with singular points P, J1, J2 and fixed points F1, F2, F'1, F'2.

If P is not an in/excenter of ABC, Psi_P(P*) lies on the circumcircle (O) : it is the isogonal conjugate of the infinite point of the line PP*. Hence, the Psi_P image of (O) is a circle C_P passing through P* analogous to the Brocard circle obtained when P = X(2). The center of C_P is the Psi_P image of the inverse of P in (O). C_P also contains the Psi_P images Ap, Bp, Cp of A, B, C which are the vertices of a triangle (analogous to the second Brocard triangle) perspective at P* to ABC. This triangle is equilateral if and only if P = X(15) or X(16).

***

A Psi_P pivotal cubic is the locus of M such that M, Psi_P(M) and a fixed point Q are collinear. This cubic is a focal cubic passing through P, J1, J2, F1, F2, F'1, F'2, Q, the infinite point of the line PQ (which is the orthic line) and Psi_P(Q) which is the singular focus F. The polar conic of F is the circle passing through F, P, Q.

When Q = P*, this focal cubic is an isogonal nK, locus of foci of inconics with center on the line PP*.