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Consider a triple {x,y,z} of real (or sometimes complex) numbers and define the point F(x,y,z) = S / √3 x X(6) + 2y X(4) + z X(3) where S = 2 area(ABC). Obviously, F(1,0,0)= X(6), F(0,1,0) = X(4), F(0,0,1) = X(3) and also F(0,1,2) = X(2), F(0,1,1) = X(5), F(3,2,1) = X(13), F(√3,0,1) = X(371), etc. Conversely, if X(n) is a center such that there exists a triple such that F(x,y,z) = X(n), then X(n) will be said to be a KHO-center and {x,y,z} will be called its KHO-coordinates. Now, let P(x,y,z) = 0 be a homogeneous polynomial equation of degree n with constant coefficients, and let C(x,y,z) = 0 be the curve (of same degree) that is the locus of the corresponding points F(x,y,z). P(x,y,z) = 0 is then called the KHO-equation of C. Conversely, if a curve has a barycentric equation that can be associated to some KHO-equation, then C will be said to be a KHO-curve. Examples : • The Euler line, Brocard axis, van Aubel line, Fermat axis, Napoleon axis are KHO-lines with KHO-equations x = 0, y = 0, z = 0, y - 2z = 0, 3y - 2z = 0 respectively. • The Kiepert hyperbola and the Evans conic are KHO-conics with KHO-equations x^2 - 3z (2y - z) = 0 and x^2 - 2 y^2 - (y - z)^2 = 0 respectively. • K369 and K458 are two nodal KHO-cubics with KHO-equations x^2 (2y + z) - 3 (2y - z) (y + z)^2 = 0 and x^2 (2y - 3 z) - 3 (y - z)^2 (2y - z) = 0 respectively. • K1191, K1192, K1193, K1194 are other KHO-cubics. See analogous cubics in the tables below. • Q073, Q167, Q168, Q183 are examples of KHO-quartics.
Most of the time, KHO-equations are far more simple than the corresponding barycentric equations. |
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KHO-equations of the form : x^2 (v y + w z) - (2y - z)(p y^2 + q z^2 + r y z) = 0 Here, v, w, p, q r are five real numbers and • v y + w z = 0 is the KHO-equation of a line L0 passing through K = X(6). • 2y - z = 0 is the KHO-equation of the line GK. • p y^2 + q z^2 + r y z = 0 is the KHO-equation of a pair of lines L1, L2, passing through K, which can be real and distinct, coincidental, imaginary depending on whether ∆ = r^2 - 4 p q is positive, null, negative. The corresponding cubic (K) has the following properties. • (K) passes through K which is a point of inflexion. The inflexional tangent is L0 and the harmonic polar is the Euler line. • (K) passes through G and meets the Euler line again at two points E1, E2 which lie on L1, L2. The nature of these two points depends on ∆.. The tangents to (K) at the three points pass through K. • When L1 = L2 ≠ L0, (K) has a node on the Euler line . See K369 and K458 for instance. • When L1 = L0 ≠ L2, (K) has a node at G. • When L1 = L2 = GK ≠ L0, (K) has a cusp at G with cuspidal tangent the Euler line. See K1195 and Table 3 below for other examples. The following table shows a selection of cubics of this type (with contributions by Peter Moses). G and K on the cubic are not repeated. The coloured cells correspond to certain types of cubics detailed below. The yellow lines are those for the Evans cubics passing through G and K. Other Evans cubics below. The grey lines are those containing the pivotal KHO-cubics described in page K1191. |
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Other Evans KHO-cubics These are the cubics passing through X(13), X(14), X(15), X(16), X(17), X(18). Those also passing through X(2) and X(6) are mentioned in the yellow lines of the previous table. Those also passing through X(2), X(3) and X(4) are in a same pencil with a KHO-equation : x (x^2 - 3 y^2 + 2 y z - z^2) + T y (x^2 - 6 y z + 3 z^2) = 0, where T is a real number or infinity, in which we recognize the KHO-equations of the Evans conic and the Kiepert hyperbola respectively : x^2 - 3 y^2 + 2 y z - z^2 = 0 and x^2 - 6 y z + 3 z^2 = 0. Another way to write the KHO-equation of these cubics is : (1 - T) (2 x - 3 y) (x + z) (x + 2 y - z) + T (2 x + 3 y) (x - z) (x - 2 y + z) = 0, where the left-hand product is the union of the lines {2,14,15}, {3,13,17}, {4,16,18}, and the right-hand product is the union of the lines {2,13,16}, {3,14,18}, {4,15,17}. Recall that X(3) is a point of inflexion and that the tangentials of X(2), X(4) are collinear with X(3) and lie on the Napoleon line and Fermat axis respectively.
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Cullen cubics : KHO-equations of the form : x^2 (v y + w z) - 3 (2y - z)(y + z)^2 = 0 See pink cells in the table above. Here, v, w are two distinct real numbers such that v + 2w ≠ 0. Every Cullen cubic (K) passes through G, K (which is a point of inflexion) and has a node at infinity, namely the infinite point X(30) of the Euler line. The polar conic (C) of X(30) has KHO-equation : (v - w) x^2 -9 (y + z)^2 = 0 showing that (K) is acnodal when v < w and crunodal when v > w. In this latter case, (K) has two real parallel asymptotes. See K369 and K1197. (C) meets the lines OK and HK at two pairs of points having KHO-coordinates (±3, 0, √(v-w)) and (±3, √(v-w), 0) respectively. These points are real when (K) is crunodal. The Hessian (H) of (K) has KHO-equation : (v - w) x^2 (2 w y + (v + w) z) + 9 (v y+w z) (y + z)^2 = 0. (K) and (H) meet at K, X(30) which is sextuple, and two other points of inflexion F1, F2 on the line with KHO-equation : (3v - 2w) y - (v -2w) z = 0. Their KHO-coordinates are (± 4√3 √(w - v), v - 2w, 3v - 2w) hence they are real when (K) is acnodal. For instance, with v = 2, w = 5, (K) is the acnodal cubic K1198 and F1, F2 are X(13), X(14). The polar conic of K splits into the tangent at K with equation v y + w z = 0 and the harmonic polar which is the Euler line. The polar conics of F1, F2 split into the respective tangents which meet at (0, 3 v - 2 w, 5 v - 6 w) on the Euler line and the harmonic polars which are parallel to the Euler line. Additional cubics computed by Peter Moses in Table 1 below. |
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KHO-equations of the form : x^2 (v y + w z) - 3 (2y - z)(y - z)^2 = 0 See green cells in the table above. Here, v, w are two real numbers such that v + w ≠ 0 and v + 2w ≠ 0. Every cubic (K) passes through G, K (which is a point of inflexion) and has a node at the nine-point center X(5) of the Euler line. The polar conic (C) of X(5) has KHO-equation : (v + w) x^2 -3 (y - z)^2 = 0 showing that (K) is acnodal when v + w < 0 and crunodal when v + w > 0. In this latter case, (K) has two real parallel asymptotes. (C) meets the line GK at two points having KHO-coordinates (±3, 0, √(v+w)). These points are real when (K) is crunodal. The Hessian (H) of (K) has KHO-equation : (v + w) x^2 [2 (2v + 3w) y - (3v + 5w) z] - 3 (v y+w z) (y - z)^2 = 0. (K) and (H) meet at K, X(5) which is sextuple, and two other points of inflexion F1, F2 on the line with KHO-equation : (7v + 6w) y - (3v +2w) z = 0. Their KHO-coordinates are (± 4 √(- w - v), 3v + 2w, 7v + 6w) hence they are real when (K) is acnodal. The polar conic of K splits into the tangent at K with equation v y + w z = 0 and the harmonic polar which is the Euler line. The polar conics of F1, F2 split into the respective tangents which meet at (0, 9v + 10w, 17v + 18w) on the Euler line and the harmonic polars which pass through X(5) and meet the line GK at (±3, √(- w - v), √(- w - v)). For instance, with v = 2, w = -3, (K) is the acnodal cubic K458 and F1, F2 are X(15), X(16). Additional cubics computed by Peter Moses in Table 2 below. |
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Tables of additional cubics Table 1 : KHO-equations : x^2 (v y + w z) - 3 (2y - z)(y + z)^2 = 0 with v - w ≠ 0 and v + 2w ≠ 0, nodal cubics with node X(30). Table 2 : KHO-equations : x^2 (v y + w z) - 3 (2y - z)(y - z)^2 = 0 with v + w ≠ 0 and v + 2w ≠ 0, nodal cubics with node X(5). Table 3 : KHO-equations : x^2 (v y + w z) - (2y - z)^3 = 0 with v y + w z ≠ 2y - z, cuspidal cubics. |
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