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Let P = p : q : r be a fixed point and M a variable point. Denote by AmBmCm the pedal triangle of M and by MaMbMc the triangle formed by the orthogonal projections of M on the cevian lines of P.

If P = H, these two triangles are always perspective and the perspector is f(P), equivalently the midpoint of the points M, H/M (H-Ceva conjugate of M) or the barycentric product of M and its orthocorrespondent.

If P is distinct of H, the triangles are perspective if and only if M lies on a circular pK with pivot P whose pole is g(P). The first barycentric coordinate of g(P) is : a^2[-2 SA p^2 + a^2(qr+rp+pq) - p(b^2q+c^2r)]. g(P) is the pole of the isoconjugation which swaps P and the inverse (in the circumcircle) of the isogonal conjugate of P.

 

Circular isogonal pKs

g maps any real point at infinity to the Lemoine point K hence there are infinitely many isogonal circular pKs. The most famous is the Neuberg cubic K001 with P = X(30) = infinite point of the Euler line. See examples in the table below and also Special Isocubics §4.1.1.

P

centers on the cubic

cubic

X30

X1, X3, X4, X13, X14, X15, X16, X30, X74, X370, X399, X484, X616, X617, X1138, X1157, X1263, X1276, X1277, X1337, X1338, X2132, X2133, etc

K001

X512

X1, X99, X512, X2142, X2143, Brocard points

K021

X513

X1, X100, X513, X1054

K661

X515

X1, X36, X40, X80, X84, X102, X515

K269

X518

X1, X105, X518, X1282

 

X522

X1, X109, X522, X1768

 

X524

X1, X2, X6, X111, X524

 

X527

X1, X9, X57, X527, X2291

 

X698

X1, X194, X698, X699

 

X736

X1, X32, X76, X736, X737

 

X744

X1, X31, X75, X744, X745

 

X1503

X1, X20, X64, X147, X1297, X1503

K270

 

Circular pKs with pivot H

The reciprocal image of H under g is the orthic axis. Hence, any circular pK with pole on the orthic axis must have its pivot at H and then, the singular focus is the antipode of the red point on the nine-point circle. The following table gives examples of such cubics. See also CL019.

Pole

centers on the cubic

cubic

X230

X4, X487, X488

 

X232

X4, X114, X371, X372, X511, X2009, X2010

K337

X468

X2, X4, X126, X193, X468, X524, X671, X2374

K209

X523

X4, X127, X525, X1289

 

X647

X4, X122, X520, X1301

 

X650

X4, X123, X521

 

X1886

X4, X118, X516, X917

 

X1990

X4, X13, X14, X30, X113, X1300

K059 = O(X4)

X2489

X4, X112, X115, X512

 

X2501

X4, X107, X125, X523

 

X3003

X3, X4, X131, X155, X265, X403, X1299, X1986

K339

 

Circular pKs with pole W ≠ K and pivot P ≠ H

For any pole W = p : q : r different of K, there is one and only one circular pK with pole W. Its pivot is P = h(W) where h is the reciprocal transformation of g. The first barycentric coordinate of P is : b^2c^2p(q+r-p) - (b^4pr+c^4rp-a^4qr). See Special Isocubics §4.2.1.

Recall that, for any pivot P = u : v : w different of H, there is one and only one circular pK with pivot P. See table 12 for a selection of such cubics with given pivot P.

Any cubic of this type is actually an isogonal pK with respect to a triangle A2B2C2 where A2 is the intersection of the lines A, gcP and Pa, P/inf where gcP is the isogonal conjugate of the complement of P (a point on the circum-conic with perspector W and on the cubic), PaPbPc is the cevian triangle of P, P/inf is the P-Ceva conjugate of the real infinite point of the cubic. Obviously, inf is the pivot of the cubic when the reference triangle is A2B2C2.

See the Droussent cubic K008 for an example of such triangle A2B2C2.

 

Circular pKs passing through a given point

A circular pK with pivot P contains

the incenter X1 if and only if P lies on the line at infinity (isogonal pKs, see above) or on the Feuerbach hyperbola. In this latter case, the pole lies on nK(X32, X513, X6).

the centroid X2 if and only if P lies on the Droussent cubic K008. The pole lies on K043.

the circumcenter X3 if and only if P lies on the circumcircle (see inversible pKs) or on the Euler line. In this latter case, the pole W must lie on the Brocard axis OK and the isoconjugate P* of the pivot is the inverse of the isogonal conjugate of P, a point on the Jerabek strophoid K039. The construction is easy since we know P and P* (see Special Isocubics §1.4.3 and §4.2.1).

The inversive image of this cubic in the circumcircle is another cubic of the same type with pivot the inverse of P in the circumcircle and pole the conjugate (on the line OK) of W in the circum-conic with perspector X(184).

For example, with P = X(30) we obtain K001 and K073, with P = X(2) we find K043 and K108.

the orthocenter X4 if and only if P lies on the Ehrmann strophoid K025. More precisely, if P is H, the pole lies on the line at infinity (see CL019). If P is not H, the pole lies on the circum-conic passing through G and K. For example, with P = G, we obtain the Droussent cubic K008.

the Lemoine point X6 if and only if P lies on K273 = pK(X111, X671). The pole lies on pK(X32<>X111, X111).

the Fermat points X13, X14 if and only if P lies on Kn = K060. The pole lies on K095.

the isodynamic points X15, X16 if and only if P lies on the Neuberg cubic K001. The pole lies on pK(X6<>X50, X6).

***

More generally, a circular pK with pivot P passes through a given point Q if and only if P lies on a circular circum-cubic K_Q with equation

a^2 v w (w y - v z) [-a^2 (x+y) (x+z) + b^2 x (x+y) + c^2 x (x+z)] = 0

which contains H, Q, agQ, giQ, the infinite point ∞Q of Q-gQ, the Ceva point CQ of Q and iQ (on the circumcircle) and also the points S, T, X, Y such that:

  • S = H-giQ /\ Q-CQ,
  • T = Q-giQ /\ agQ-CQ,
  • X second intersection of the line Q-gQ with the rectangular circum-hyperbola passing through gQ and giQ, isogonal transform of the line OQ,
  • Y second intersection of the line H-∞Q with the rectangular circum-hyperbola passing through CQ.

The pole also lies on a circum-cubic Ω_Q passing through K, Q x igQ and more generally the poles of all the pKs with pivots given above.

Furthermore,

  • K_Q has a flex at Q if and only if Q lies on K028, the Musselman (third) cubic.
  • the polar conics of X(5) and ∞Q in K_Q are always rectangular hyperbolas. In other words, the orthic line of K_Q (and K_gQ) is the line X(5)-∞Q.
  • thus, the polar conic of Q in K_Q is a rectangular hyperbola if and only if Q lies on K005, the Napoleon cubic. In this case, K_Q is a focal cubic.
  • K_Q is a K0 if and only if Q lies on K018, the Brocard (second) cubic, and in this case, it is a pK.

***

When Q lies on the circumcircle, K_Q is psK(Q, tgQ, H) therefore it is tangent at A, B, C to the symmedians. Hence, all these cubics form a pencil and the singular focus lies on the circle C(O, 2R). K_Q meets the sidelines of ABC at the vertices of the cevian triangle of tgQ, the isotomic conjugate of the isogonal conjugate of Q, a point on the Steiner ellipse. The tangent at Q passes through O. The infinite real point of K_Q is the isogonal conjugate of Q and the real asymptote envelopes a deltoid, the homothetic of the Steiner deltoid under h(G,4). See the figure and the table below.

Q

points on the cubic K_Q

cubic

remark

X(74)

X(3), X(4), X(30), X(74)

 

 

X(98)

X(4), X(76), X(98), X(511)

 

 

X(104)

X(4), X(8), X(104), X(517)

 

 

X(106)

X(1), X(4), X(106), X(519), X(1168), X(1320)

 

 

X(107)

X(4), X(107), X(250), X(520)

 

 

X(108)

X(4), X(59), X(108), X(521)

 

 

X(111)

X(2), X(4), X(6), X(23), X(111), X(524), X(671), X(895)

K273

pK

X(112)

X(4), X(112), X(249), X(525)

 

 

X(733)

X(4), X(83), X(732), X(733), X(1916)

 

 

X(759)

X(4), X(21), X(58), X(80), X(758), X(759), X(1325)

K306

 

X(1141)

X(4), X(54), X(265), X(1141), X(1154)

 

 

X(2291)

X(4), X(9), X(57), X(527), X(1156), X(2291)

 

 

X(2373)

X(4), X(69), X(316), X(1177), X(2373), X(2393)

 

 

X(2380)

X(4), X(14), X(15), X(532), X(1337), X(2380)

 

 

X(2381)

X(4), X(13), X(16), X(533), X(1338), X(2381)

 

 

***

When Q lies on the line at infinity, K_Q contains the vertices Ga, Gb, Gc of the antimedial triangle. Hence, all these cubics form a pencil and the singular focus lies on the circle C(H, R).

Q

points on the cubic K_Q

cubic

remark

X(30)

X(4), X(20), X(30), X(146), X(265), X(1138), X(1294)

K449

 

X(511)

X(4), X(147), X(315), X(511), X(1297), X(1670), X(1671), X(1916)

 

 

X(514)

X(4), X(150), X(514), X(1305)

 

 

X(517)

X(4), X(153), X(517), X(1295), X(1320)

 

 

X(519)

X(4), X(8), X(80), X(519), X(2370)

 

 

X(523)

X(4), X(523), X(925)

 

focus X(265)

X(524)

X(2), X(4), X(67), X(69), X(316), X(524), X(671), X(858), X(2373)

K008

pK

X(527)

X(4), X(7), X(329), X(527)

 

 

X(532)

X(4), X(13), X(532), X(616), X(622)

 

 

X(533)

X(4), X(14), X(533), X(617), X(621)

 

 

X(1154)

X(3), X(4), X(1154), X(3153)

 

 

X(2393)

X(4), X(23), X(66), X(895), X(1370), X(2393)

 

 

***

K_Q and K_gQ generate a pencil of circular cubics passing through H and meeting the line at infinity at the same real point S, namely that of the line Q-gQ. This pencil contains :

• pK(H x S, H) where H x S is a point on the orthic axis, see above,

• the decomposed cubic which is the union of the line at infinity and the rectangular circum-hyperbola H_Q passing through the isogonal conjugate of the NK-Transform of Q. Recall that the NK-Transform of Q is the pole of the line Q-gQ in the circum-conic passing through Q and gQ.

It follows that all these cubics must have two other common points. These points are the intersections of the polar of S in H_Q with H_Q.

CL035faisceau