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The tripolar centroid TG(M) of a point M (distinct of G) is the isobarycentre of the traces of the trilinear polar of M on the sidelines of ABC. This notion was introduced by Darij Grinberg and Wilson Stothers has written an excellent note on this subject available here.

Let L be a line whose trilinear pole is Q = p:q:r.

Tripolar centroidal pre-image of a line

The locus of M such that TG(M) lies on L is a circumcubic K0(Q) with equation :

∑ q r x (y + z - 2x)(y - z) = 0.

K0(Q) is a nodal cubic with node G. The nodal tangents are parallel to the asymptotes of the circumconic passing through G and Q.

Since K0(Q) is unicursal, it is fairly easy to parametrize it : for any point X in the plane distinct of G, the point X' lies on the cubic. With X = x:y:z, we have :

X' = p [q(x + y - 2z) - r(x - 2y + z)] / (y - z) : : .

K0(Q) also contains :

  • Q,
  • S = 2p(q + r) - qr : : , the reflection in G of the isotomic conjugate tQ of Q,
  • T intersection of the line QS and the parallel at G to L,
  • the third point on BC which is U = 0 : 2q + r : q + 2r, on the parallel at S to A tQ, V and W similarly,
  • the third point on AQ which is U' = q + r : 2q : 2r, on this same parallel, V' and W' similarly.

Now, if P = TG(M), there is another point M' such that TG(M') = P. M' lies on the circumconic with perspector P and on the parallel at TG(P) to the trilinear polar of tP. M and M' coincide if and only if M lies on K015, the Tucker nodal cubic.

If M = u:v:w, this point M' is (v + w - 2u) / (uv + uw - 2 vw) : : . We shall say that M' is the Tripolar Centroidal Conjugate of M and write M' = TCC(M).

The mapping M -> M' is an involution of degree 5, with singular points A, B, C, G (thrice) and fixed points those of K015. It follows that the transform of a line passing through G (which is not a median) is a circum-conic also passing through G. For example, the line X(2)X(523) gives the Kiepert hyperbola and the line X(2)X(512) gives the circum-conic passing through X(6).

When M lies on K0(Q), M' obviously also lies on the cubic and the line MM' passes through S. This is in particular the case of Q and T.

Special cases

The cubic K0(Q)

  • cannot be a proper pK since the triangles ABC and UVW are perspective when Q lies on a median in which case the cubic decomposes.
  • is a nK0 if and only if Q lies at infinity (in which case the cubic decomposes) or on the Steiner ellipse (in which case the cubic is an isotomic nK0 with root at infinity, a member of the class CL031).
  • has perpendicular nodal tangents if and only if Q lies on the Kiepert hyperbola. See K185, K295 for example.
  • is circular if and only if Q = X(67). This is K394.
  • is equilateral (a K60 cubic) if and only if Q = X(11058). This is K395.


Tripolar centroidal image of a line

When M traverses L, the locus of TG(M) is another nodal cubic with node S, the reflection of tQ about G, but it is not a circumcubic.

It has three real asymptotes parallel to the sidelines of ABC and it meets these sidelines again at six real points that lie on a same conic C(Q) with center the midpoint of G-ctQ or equivalently the homothetic of S under h(G, 1/4).

The points on BC are U as above and U1 = 0 : q(2p+r) : r(2p+q).

When Q = G, L is the line at infinity and U, U1 coincide at the midpoint of BC. C(G) is the inscribed Steiner ellipse and the cubic is K219.

When Q = H, L is the orthic axis and C(Q) is the circle with center the midpoint X(597) of GK passing through X(115) and X(125), the centers of the Kiepert and Jerabek hyperbolas. The cubic has a node at X(1992). See figure below.