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These cubics are introduced by Clark Kimberling and Peter Moses in the preamble just before X(36810) in ETC as follows.

Let P = p : q : r and U = u : v : w be points not on the sidelines BC, CA, AB of a triangle ABC.

Let A'B'C' and A"B"C" be the cevian triangles of P, U respectively and A* be the midpoint of A' and A''. Define B* and C* cyclically.

The triangle A*B*C* is named the mid-trace triangle of P and U, denoted by M(P,U)

For given P, the locus of a point X = x : y : z such that M(P,X) is the cevian triangle of some point Q is given by the cubic

p (q r + r^2 + q p) y^2 z - p (q r + q^2 + r p) y z^2 + (cyclic) + (p - q) (p - r) (q - r) x y z = 0,

named the mid-cevian cubic of P, denoted by MC(P).

When X varies on MC(P), the locus of Q is a very similar cubic MC'(P) with equation

p (p q+3 p r+q r+r^2) y^2 z - p (3 p q+q^2+p r+q r) y z^2+ (cyclic) +(p - q) (p - r) (q - r) x y z = 0.

***

Special cases (excluded in the sequel)

• If P is on the line at infinity, then MC(P) is the union of the line at infinity and the circumconic passing through P and its isotomic conjugate tP.

• If P is on the Steiner ellipse, then MC(P) is the union of the line through P, tP and the Steiner ellipse.

• If P = X(2), then MC(P) is the union of the medians of ABC.

***

Another characterization and a construction of MC(P)

Let (L) be a line passing through tP, (L') its parallel at P, (C) its isotomic transform. When (L) rotates about tP, (L') and (C) meet at two points on MC(P). This is very similar to the spK cubics of CL055.

It follows that a variable line passing through P meets the circum-conic passing through P and the isotomic transform of the infinite point of the line at P and another point on MC(P).

***

Properties of MC(P)

MC(P) is a nodal cubic with node P. The nodal tangents are parallel to the asymptotes of the circumconic C(P) passing through P and tP, the isotomic transform of the line P-tP.

MC(P) meets the line at infinity at the same points as pK(X2, tP) and the Steiner ellipse at the same points as pK(X2, P).

The isotomic transform of MC(P) is MC(tP).

If A1B1C1 is the circumcevian triangle of P in the Steiner ellipse, then the lines passing through tP and A1, B1, C1 meet BC, CA, AB at U, V, W on MC(P).

MC(P) is a circular cubic if and only if pK(X2, tP) is the Droussent cubic K008 = pK(X2, X316) hence P must be X(67).

MC(P) is an equilateral cubic if and only if pK(X2, tP) is K092 = pK(X2, X11057) hence P must be X(11058).

MC(P) is a psK if and only if P lies on K327 and a nK if and only if P lies on K016. These are two Tucker cubics as in CL064.

The nodal tangents at P are perpendicular if and only if C(P) is a rectangular hyperbola hence passing through H. It follows that P must lie on the Lucas cubic K007.

P is a cusp on MC(P) if and only if C(P) is a double line hence P must lie on K015, another Tucker cubic. In such case, tP also lies on K015 and the line P-tP is tangent to the Steiner ellipse which is the pivotal conic of K015. The other tangent at P to the Steiner ellipse is the cuspidal tangent.

***

Points on MC(P)

Apart the node P, MC(P) contains the following points :

tP = q r : : , isotomic conjugate of P,

P' = 2 q+2 r-p : : , reflection of P in G,

G/P = p (p-q-r) : : , G-Ceva conjugate of P,

G©P = p (p q-p r-q r) (p q-p r+q r) : : , crossconjugate of G and P,

P1 = (p q-2 p r-2 q r) (2 p q-p r+2 q r) : : , isotomic conjugate of the reflection of tP in G,

P2 = (2 p-q-r) (2 p q-p r-q r) (p q-2 p r+q r) : : , barycentric quotient of the infinite points of the lines GP and GtP,

P3 = (p^2+q^2-p r-q r) (-p q-p r+2 q r) (-p^2+p q+q r-r^2) : : ,

P4 = (p^2-q r) (p^2-p q+q^2-r^2) (p^2-q^2-p r+r^2) : : ,

P5 = p (-q^2+p r) (p q-r^2) (p^2 q^2-p^2 q r+p^2 r^2-q^2 r^2) : : ,

P6 = p (p+q-2 r) (p-2 q+r) (p q^2-q^2 r+p r^2-q r^2) : : ,

P7 = p (p^2 q+p q^2-p^2 r-q^2 r) (p^2 q-p^2 r-p r^2+q r^2) (p^2 q^3-p^2 q^2 r-p^2 q r^2+2 p q^2 r^2-q^3 r^2+p^2 r^3-q^2 r^3) : : ,

P8 = p (p q-q^2+p r-r^2) (p^3-p^2 q-p q^2+q^3+2 p q r-p r^2-q r^2) (p^3-p q^2-p^2 r+2 p q r-q^2 r-p r^2+r^3) : : ,

P9 = p (2 p q-p r-q r) (p q-2 p r+q r) (2 p^2 q^3-p^2 q^2 r-p^2 q r^2+2 p q^2 r^2-2 q^3 r^2+2 p^2 r^3-2 q^2 r^3) : : ,

P10 = p (p^2 q+p q^2-2 p^2 r-2 q^2 r+p r^2+q r^2) (2 p^2 q-p q^2-p^2 r-q^2 r-p r^2+2 q r^2) (p^2 q^3-2 p^2 q^2 r-2 p^2 q r^2+4 p q^2 r^2-q^3 r^2+p^2 r^3-q^2 r^3) : : ,

P11 = p (p^3-p q^2-q^2 r-p r^2-q r^2) : : ,

P12 = p (p+q-r) (p-q+r) (p^3 q-p q^3+p^3 r-p^2 q r+p q^2 r-q^3 r+p q r^2-2 q^2 r^2-p r^3-q r^3) : : ,

P13 = (2 p-q-r) (2 p^3-p^2 q-p q^2+2 q^3+2 p q r-2 p r^2-2 q r^2) (2 p^3-2 p q^2-p^2 r+2 p q r-2 q^2 r-p r^2+2 r^3) : : ,

P14 = (p^3-2 p^2 q-2 p q^2+q^3+4 p q r-p r^2-q r^2) (p^2 q-2 p q^2+p^2 r+q^2 r-2 p r^2+q r^2) (p^3-p q^2-2 p^2 r+4 p q r-q^2 r-2 p r^2+r^3) : : ,

P15 = p^2 (p^2 q^2+p q^3+q^3 r-p^2 r^2+q^2 r^2) (p^2 q^2-p^2 r^2-q^2 r^2-p r^3-q r^3) : : ,

P16 = p (p q+p r-q r) (p^3 q^2+p^2 q^3-p^2 q^2 r-p^3 r^2+p^2 q r^2+p q^2 r^2-q^3 r^2-p^2 r^3-2 p q r^3-q^2 r^3) (p^3 q^2+p^2 q^3-p^2 q^2 r+2 p q^3 r-p^3 r^2+p^2 q r^2-p q^2 r^2+q^3 r^2-p^2 r^3+q^2 r^3) : : ,

with these triads of collinear points on MC(P) :

{tP, P', P6} – {tP, G/P, P3} – {tP, G©P, P7} – {tP, P1, P9} – {tP, P2, P10} – {tP, P4, P11} – {tP, P8, P12} – {P', G/P, P4} – {P', P1, P5} –

{P', P7, P15} – {P', P11, P13} – {G/P, G©P, P5} – {G/P, P8, P11} – {G/P, P9, P15} – {G©P, P4, P12} – {G©P, P6, P11} – {P1, P3, P11} –

{P2, P11, P14} – {P3, P12, P15} – {P5, P11, P15} – {P5, P12, P16} – {P7, P11, P16}.

Now, if M = f(p:q:r) : : is a point on MC(P), then the isotomic conjugate N of the point obtained from M under the substitution SS{p -> 1/p} is another point on MC(P). M and N are not always distinct.

M

P

tP

G/P

P'

P2

P3

P4

P7

P9

P10

P11

P12

 

N

P

tP

G©P

P1

P2

P6

P5

P8

P13

P14

P15

P16

 

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The following table presents a selection of these cubics MC(P) computed by Peter Moses.

P

MC(P) passes through A, B, C and X(i) for these i

notes or other points on MC(P)

X(1)

1, 9, 75, 87, 993, 3679, 24806, 36814, 36815, 36816, 36817, 36818, 36819

 

X(4)

4, 69, 376, 1249, 3421, 5485, 6601, 9214, 34208

MC(X4) is K616

X(6)

3, 6, 76, 599, 3224, 9462, 14608, 19127, 36820, 36821, 36822, 36823

vertices of the Brocard triangle

X(7)

7, 8, 3160, 6172, 27818, 36588

 

X(67)

67, 265, 316, 524, 8724, 11646, 15900, 34319, 36824, 36825, 36825, 36826

circular points at infinity

X(69)

4, 69, 1992, 6337, 6604, 34403

 

X(74)

74, 477, 895, 3260, 5627, 10706

 

X(76)

6, 76, 264, 598, 6374, 7757

 

X(98)

98, 325, 842, 5503, 6054, 16092

 

X(100)

100, 693, 1290, 4767, 5375, 10707

 

X(110)

110, 476, 850, 9140, 9146, 17708, 27867, 36827, 36828, 36829, 36830, 36831

 

X(476)

110, 476, 3268, 9140, 17708, 34312

 

X(850)

110, 850, 3268, 8599, 9141, 9979

 

 

 

 

The yellow cells correspond to points on K007 hence MC(P) has perpendicular nodal tangents.