User:Clark Kimberling
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[edit] Cubic curves in the plane of a triangle
Suppose that ABC is a triangle with sidelengths a = |BC|, b = |CA|, c = |AB|. Relative to ABC, many named cubics pass through well known points. Examples shown below using two different kinds of homogeneous coordinates: trilinear and barycentric.
To convert from trilinear to barycentric in a cubic equation, substitute as follows:
x -> bcx, y -> cay, z -> abz;
to convert from barycentric to trilinear, use
x -> ax, y -> by, z -> cz.
Many equations for cubics have the form
f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,z,x,y) = 0.
In the examples below, such equations are written more succinctly in "cyclic sum notation", like this:
[cyclic sum f(x,y,z,a,b,c)] = 0.
The cubics listed below can be defined in terms of the isogonal conjugate, denoted by P*, of a point P not on a sideline of ABC. A construction of P* follows: Let LA be the reflection of line PA about the internal angle bisector of angle A, and define LB and LC analogously. Then then three reflected lines concur in P*. Using trilinear coordinates, if P = p:q:r, then P* = 1/p:1/q:1/r.
[edit] Neuberg cubic
Trilinear equation: [cyclic sum (cos A - 2cos B cos C)x(y2 - z2)] = 0
Barycentric equation: [cyclic sum (a2(b2 - c2) - (b2 - c2 - 2a4)2)x(c2y2 - b2z2)] = 0
The Neuberg cubic is the locus of a point X such that X* is on the line UX, where X is the Euler infinity point (X30 in the Encyclopedia of Triangle Centers). Also, this cubic is the locus of X such that the triangle XAXBXC is perspective to ABC, where XAXBXC is the reflection of X in the lines BC, CA, AB, respectively
The Neuberg cubic passes through the following points: incenter, circumcenter, orthocenter, both Fermat points, both isodynamic points, the Euler infinity point, other triangle centers, the excenters, the reflections of A, B, C in the sidelines of ABC, and the vertices of the six equilateral triangles erected on the sides of ABC.
For a graphical representation and extensive list of properties of the Neuberg cubic, see K001 at Berhard Gibert's Cubics in the Triangle Plane.
[edit] Thomson cubic
Trilinear equation: [cyclic sum bcx(y2 - z2)] = 0
Barycentric equation: [cyclic sum x(c2y2 - b2z2)] = 0
The Thomson cubic is the locus of a point X* is on the line UX, where X is the centroid.
The Thomson cubic passes through the following points: incenter, centroid, circumcenter, orthocenter, symmedian point, other triangle centers, the vertices A, B, C, the excenters, the midpoints of sides BC, CA, AB, and the midpoints of the altitudes of ABC. For each point P on the cubic but not on a sideline of the cubic, the isogonal conjugate of P is also on the cubic.
For a graphical representation and list of properties of the Thomson cubic, see K002 at Cubics in the Triangle Plane.
[edit] Darboux cubic
Trilinear equation: [cyclic sum (cos A - cos B cos C)x(y2 - z2)] = 0
Barycentric equation: [cyclic sum (2a2(b2 + c2) + (b2 - c2)2 - 3a4)x(c2y2 - b2z2)] = 0
The Darboux cubic is the locus of a point X* is on the line UX, where X is the de Longchamps point, (X20 in the Encyclopedia of Triangle Centers). (The point lies on the Darboux cubic.) Also, this cubic is the locus of X such that the pedal triangle of X is the cevian of some point. (The point lies on the Lucas cubic.) Also, this cubic is the locus of a point X such that the pedal triangle of X and the anticevian triangle of X are perspective. (The perspector lies on the Thomson cubic.)
The Darboux cubic passes through the incenter, circumcenter, orthocenter, de Longchamps point, other triangle centers, the vertices A, B, C, the excenters, and the antipodes of A, B, C on the circumcircle. For each point P on the cubic but not on a sideline of the cubic, the isogonal conjugate of P is also on the cubic.
For a graphics and properties, see K004 at Cubics in the Triangle Plane.
[edit] Napoleon-Feuerbach cubic
Trilinear equation: [cyclic sum cos(B - C)x(y2 - z2)] = 0
Barycentric equation: [cyclic sum a2(b2 - c2) - (b2 - c2)2)x(c2y2 - b2z2)] = 0
The Napoleon-Feuerbach cubic is the locus of a point X* is on the line UX, where X is the nine-point center, (X5 in the Encyclopedia of Triangle Centers).
The Napoleon-Feuerbach cubic passes through the incenter, circumcenter, orthocenter, 1st and 2nd Napoleon points, other triangle centers, the vertices A, B, C, the excenters, the projections of the centroid on the altitudes, and the centers of the 6 equilateral triangles erected on the sides of ABC.
For a graphics and properties, see K005 at Cubics in the Triangle Plane.
[edit] Lucas cubic
Trilinear equation: [cyclic sum (cos A)x(b2y2 - c2z2)] = 0
Barycentric equation: [cyclic sum b2 + c2 - a2)x(y2 - z2)] = 0
The Lucas cubic is the locus of a point X such that the cevian triangle is the pedal triangle of some point. (The point lies on the Darboux cubic.)
The Lucas cubic passes through the centroid, orthocenter, Gergonne point, Nagel point, de Longchamps point, other triangle centers, the vertices of the anticomplementary triangle, and the foci of the Steiner circumellipse.
For a graphics and properties, see K007 at Cubics in the Triangle Plane.
[edit] 1st Brocard cubic
Trilinear equation: [cyclic sum bc(a4 - b2c2)x(y2 + z2] = 0
Barycentric equation: [cyclic sum (a4 - b2c2)x(c2y2 + b2z2] = 0
Let A'B'C' be the 1st Brocard triangle. For arbitrary point X, let XA, XB, XC be the intersections of the lines XA' , XB' , XC' with the sidelines BC, CA, AB, respectively. The 1st Brocard cubic is the locus of X for which the points XA, XB, XC are collinear.
The 1st Brocard cubic passes through the centroid, symmedian point, Steiner point, other triangle centers, and the vertices of the 1st and 3rd Brocard triangles.
For a graphics and properties, see K017 at Cubics in the Triangle Plane.
[edit] 2nd Brocard cubic
Trilinear equation: [cyclic sum bc(b2 - c2)x(y2 + z2] = 0
Barycentric equation: [cyclic sum (b2 - c2)x(c2y2 + b2z2] = 0
The 2nd Brocard cubic is the locus of a point X for which the pole of the line XX* in the circumconic through X and X* lies on the line of the circumcenter and the symmedian point (i.e., the Brocard axis).
The 2nd Brocard cubic passes through the centroid, symmedian point, both Fermat points, both isodynamic points, the Parry point, other triangle centers, and the vertices of the 2nd and 4th Brocard triangles.
For a graphics and properties, see K018 at Cubics in the Triangle Plane.
[edit] 1st equal areas cubic
Trilinear equation: [cyclic sum a(b2 - c2)x(y2 - z2] = 0
Barycentric equation: [cyclic sum a2(b2 - c2)x(c2y2 - b2z2] = 0
The 1st equal areas cubic is the locus of a point X such that area of the cevian triangle of X equals the area of the cevian triangle of X*. Also, this cubic is the locus of X for which X* is on the line S*X, where S is the Steiner point.
The 1st equal areas cubic passes through the incenter, Steiner point, other triangle centers, the 1st and 2nd Brocard points, and the excenters.
For a graphics and properties, see K021 at Cubics in the Triangle Plane.
[edit] 2nd equal areas cubic
Trilinear equation: (bz+cx)(ay+cx)(ay+bz) = (bx+cy)(az+cy)(az+bx)
Barycentric equation: [cyclic sum a(a2 - bc)x(c3y2 - b3z2] = 0
For any point X = x:y:z (trilinears), let XY = y:z:x and XZ = z:x:y. The 2nd equal areas cubic is the locus of X such that the area of the cevian triangle of XY equals the area of the cevian triangle of XZ.
The 2nd equal areas cubic passes through the incenter, centroid, symmedian point, and points in Encyclopedia of Triangle Centers indexed as X(31), X(105), X(238), X(292), X(365), X(672), X(1453), X(1931), X(2053), and others.
For a graphics and properties, see K155 at Cubics in the Triangle Plane.
