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X(4), X(5), X(6), X(15), X(16), X(30), X(19106), X(19107)

X(43622) → X(43629)

other points below

Geometric properties :

K1231 is a KHO-cubic, see K1191 for explanations and also CL075.

Its KHO-equation is (1) : x^2 (2y + z) + z (y^2 - z^2) = 0 or (2) : z (x^2 + y^2 - z^2) + 2 x^2 y = 0.

This equation can also be written as (x^2 - 3y^2 + 2yz - z^2) (2y + z) + 6y^3 = 0, showing that K1231 has a triple contact with the Evans conic at X(15) and X(16). Note that the polar conic (C) of X(5) meets K1231 again at X(15), X(16), X(19106), X(19107).

X(6) is a point of inflexion with tangent passing through X(20) and harmonic polar the Euler line. The parallel at X(6) to the Euler line is an asymptote.

K1231 is bitangent at X(5) and X(30) to the hyperbola (H) – see below – with KHO-equation x^2 + y^2 - z^2 = 0, the remaining common points being X(15) and X(16).


Other KHO-points on K1231

Q1 = (√6,2,4) and Q2 = (-√6,2,4), on the line X(2)X(6),

Q3 = (√6,-5,4) and Q4 = (√6,5,-4), on the line X(6)X(33703),

Q5 = (2√6,-1,3) and Q6 = (2√6,1,-3), on the line X(6)X(550),

Q7 = (2√6,-15,3) and Q8 = (2√6,15,-3), on the line X(6)X(3853).

Note that Q1, Q4, Q6, Q7 on the one hand and Q2, Q3, Q5, Q8 on the other, share the same tangential.


The center Ω of (H) is X(43618), intersection of the lines {4,187}, {6,30} and many others.

Its KHO-coordinates are : (cotω / √3,1,-1).


The Pythagorean hyperbola (H)

(H) passes through X(i) for these i : 5, 15, 16, 30, 590, 615, 9996, 37832, 37835, 42191, 42192, 42193, 42194, 42258, 42259, 42627, 42628, 43240, 43241, 43630 → 43649.

Every KHO-point (u,v,w) such that u^2 + v^2 = w^2 clearly lies on (H) and actually generates a group G(t) of eight points (when u v w ≠ 0) on (H) by swapping the first two coordinates and then, by sucessively changing one sign in the coordinates.

Example : X(590) = (√3,1,2) gives X(615), X(42258), X(42259) and X(42194) = (1,√3,2) gives X(42191), X(42192), X(42193).

If u = 0, the group is reduced to four members, namely X(5), X(15), X(16), X(30).

With (u,v,w) = (3,4,5) corresponding to X(37832), we obtain X(37835), X(43632), X(43633) and X(42627), X(42628), X(43630), X(43631). See figure below.

A KHO-parametrization of (H) is given by P(t) = (1 - t^2, 2t, 1 + t^2) or even (cosθ, sinθ, 1).

The 8 points of G(t) – when all are distinct – give 28 lines and 5 groups of 4 lines pass through X(3), X(4), X(6), X(5318), X(5321) respectively.

The remaining 8 lines are tangent to a same hyperbola (H') with KHO-equation : 2 x^2 + 2 y^2 - z^2 = 0 and center Ω' = (cotω / √3,1,-2).


The first four members of G(t) are therefore P(t), P(1/t), P(-t), P(-1/t) and the last four members of G(t) are P(T), P(1/T), P(-T), P(-1/T) where T = (1 - t) / (1 + t).

Two points P(t), P(t') such that t + t' = 0 are said to be opposite and collinear with X(4).

Two points P(t), P(t') such that t t' = 1 are said to be inverse and collinear with X(6).

Two points P(t), P(t') such that t t' = -1 are said to be inverse-opposite and collinear with X(3).

More generally, with s = t + t' and p = t t' , three points P(t), P(t') and the KHO-point (u,v,w) are collinear if and only if v s - (u+w) p +(u-w) = 0.

In particular, when s is constant, (u,v,w) lies on the line X(4)X(16), and when p is constant, (u,v,w) lies on the Brocard axis, in both cases for every pair of numbers {t, t'}.


Pythagorean triples of integer KHO-points on (H)

X(37832) is the KHO-point (3,4,5), also P(1/2), associated to the group of 8 points G(1/2).

It is the basis of three new Pythagorean KHO-points, namely P(2/5), P(2/3), P(1/4), hence to three groups of eight points G(2/5), G(2/3), G(1/4), that is in total 24 new integer KHO-points on (H).

This can be repeated as long as we wish and a Pythagorean KHO-point P(t) will give the three new Pythagorean KHO-points P(1 / (2+t)), P(1 / (2-t)), P(t / (1+2t)) and then, three new associated groups.

Naturally, the same process remains valid if the starting point (3,4,5) is replaced by another non-integer point such as X(590) = (√3,1,2).


In the figure opposite, t = 2.

The points P(t), P(1/t), P(-t), P(-1/t) are labelled Ro, Ra, Rb, Rc.

The points P(T), P(1/T), P(-T), P(-1/T) are labelled So, Sa, Sb, Sc.

These are the points associated to the Pythagorean triple (3,4,5). See above.

X(5318) and X(5321) are the KHO-points (1,1,0) and (-1,1,0).

X(18581) and X(42086) are (-1,1,1) and (1,1,-1).