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K1234

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X(2), X(3), X(6), X(11542), X(11543), X(42912), X(42913), X(42942), X(42943)

X(43869) → X(43889)

KHO-points (±2√2,-1,2) and (±2√6,1,6) on the KHO-hyperbola (H), see preamble just before X(43457) in ETC and further details below

Geometric properties :

K1234 is a crunodal KHO-cubic, see K1191 for explanations and CL075. See also K1233, a very similar cubic.

Its KHO-equation is equivalently

(1) : x^2 (4y - z) - 12 y^2 (2y - z) = 0

(2) : 3 (2y - z)^3 - (4y - z) (x^2 - 6 y z +3 z^2) = 0, where x^2 - 6 y z +3 z^2 = 0 is the Kiepert hyperbola,

(3) : (4y - z) (x^2 + 2 y z - z^2) - (2y - z) (2y + z) (6y - z) = 0, where x^2 + 2 y z - z^2 = 0 is the KHO-hyperbola (H), see below.

X(3) is a node with two real nodal tangents passing through X(1587) and X(1588).

X(6) is a point of inflexion with tangent passing through X(631) and harmonic polar the Euler line. Note that the polar conic of X(631) splits into the Brocard axis and the line {5,6}. This latter line meets the cubic again at X(11542), X(11543) with KHO-coordinates (±2,1,1).

X(2) is a sextactic point with tangent passing through X(6) and sextactic conic the Kiepert hyperbola.

Parametrization :

For any real (sometimes complex) number t or infinity (giving X2), the KHO-point P(t) = (12 t^2 - 1, t (12 t^2 - 1), 4 t (6 t^2 - 1)) lies on K1234. This gives a lot of simple points on the curve. Note that P(t), P(-t) and X(6) = P(0) are collinear for every t.

More generally, three points P(t1), P(t2),P(t3) are collinear on the cubic if and only if t1 + t2 + t3 + 12 t1 t2 t3 = 0.

Taking t3 = ∞, we see that P(t1), P(t2), X(2) are collinear if and only if 1 + 12 t1 t2 = 0.

Taking t1 = t2 = t, we see that the tangential of P(t) is P(T) where T = - 2t / (1 + 12 t^2).

It easily follows that P(t1) ≠ P(t2) share the same tangential if and only if 1 - 12 t1 t2 = 0.

Taking t1 = t2 = t3 = t, we obtain 3t (1 + 4 t^2) = 0, giving the three points of inflexion, namely X(6) = P(0) and two imaginary points (±4i,2,5).

***

The KHO-hyperbola (H) :

(H) passes through X(i) for these i : 2, 4, 15, 16, 316, 1348, 1349, 2039, 2040, 3096, 5418, 5420, 6560, 6561, 7790, 7859, 9993, 14165, 36969, 36970, 38227, 42936, 42937, 43195, 43196, and also the 67 points X(43460)-X(43526).

Its center is X(43457) with KHO-coordinates (cotω / √3,2,1), its perspector is X(43458) whose isotomic conjugate is X(43459).

(H) is bitangent at X(2), X(4) to the Kiepert hyperbola with common tangents passing through X(6). It is also bitangent at X(15), X(16) to the Evans conic with common tangents passing through X(5).

For any real (sometimes complex) number t or infinity (giving X2), the KHO-point H(t) = (2t, t^2 - 1, 2t^2) lies on (H).

Two points H(t) and H(t') such that

• t + t' = 0 are said to be opposite on (H) and then collinear with X(6),

• t × t' = 1 are said to be inverse on (H) and then collinear with X(5),

• t × t' = -1 are said to be inverse-opposite on (H) and then collinear with X(3).

Hence, for any t ≠ 0 and t ≠ ∞, the four points H(t), H(-t), H(1/t), H(-1/t) have the same diagonal triangle with vertices X(3), X(5), X(6).

The KHO-hyperbola (H') :

(H'), with equation x^2 - 2 y z - z^2 = 0, is closely related to (H) since it is the locus of the KHO-point H'(t) = (2t, 1 - t^2, 2t^2), so that each point (u,v,w) on (H) gives the point (u,-v,w) on (H').

It is easy to see that (H') is in fact the image of (H) under the harmonic homology having center H and axis the Brocard axis.

(H') passes through X(i) for these i : 4, 15, 16, 20, 485, 486, 9873, 42260, 42261, 42813, 42814. Its center is X(187).

The properties of two points H'(t) and H'(t') on (H'), analogous to those above for (H), are the same but X(5) must be replaced with X(30).

 

Generalized KHO-hyperbolas

All points and equations are given in KHO-coordinates unless otherwise specified.

Let k ≠ 0 be a real number and denote by (Hk) the hyperbola with equation x^2 - z^2 - k y z = 0, the locus of point Hk(t) = (k t, 1 - t^2, k t^2), where t is a real parameter or infinity.

The center Ωk of (Hk) is (- cotω / √3 k^2,2(k - 2),2 k), which lies on the ellipse √3 x (y - z) - cot2ω z^2 = 0. This ellipse has center the midpoint of X(4)-X(187), two points on the curve which also contains X(6), X(7755), X(37689). The tangent at X(4) is the Euler line and the tangent at X(187) is parallel. The tangent at X(6) passes through X(5).

(Hk) passes through

• X(4) = Hk(0) with tangent passing through X(6).

• X(15) = Hk(1) with tangent 2(x - z) - k y = 0 and X(16) = Hk(-1) with tangent 2(x + z) + k y = 0. These two tangents meet at (0,2,-k) on the Euler line.

• a second point on the Euler line which is P(∞) = (0,1,-k) with tangent also passing through X(6).

• two points E1, E2 on the Evans conic which are always real and distinct. These are the points (±√3 √[(k + 1)^2 + 2],k + 2, 3), on the line passing through X(6) and (0,2 + k,3) on the Euler line.

• two points K1, K2 on the Kiepert hyperbola which are real and distinct if and only if -2 < k < 6. These are the points (±√3 √[2 + k)(6 - k)],4, 6 - k), on the line passing through X(6) and (0,4,6 - k) on the Euler line.

Special cases

• with k = -2 and k = 2, we find (H) and (H') as above.

• with k = 6, (Hk) has a quadruple contact at X(4) to the Kiepert hyperbola. It center is X(7755).

• with k = 4, (Hk) has center X(37689) and passes through X(i) for these i : 4, 13, 14, 15, 16, 376, 3068, 3069, 5667, 9862, 42570, 42571, 42637, 42638, 42805, 42806, 42926, 42927, 42986, 42987, 43016, 43017, 43453.

• with k = -1/2, (Hk) passes through X(i) for these i : 4, 15, 16, 381, 590, 615, 42107, 42110, 42940, 42941. Its center is (- cotω / √3/ 4,5,1), on the line {6,1327}.