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X(1), X(2), X(3), X(40), X(376), X(5646), X(14482), X(46943), X(46944), X(46945), X(46946), X(46947)

excenters and their reflections in O

vertices of the Thomson triangle Q1Q2Q3

infinite points of pK(X6, X10304)

other points below

Geometric properties :

K1262 is K047 for the Thomson triangle. Recall that K047 is spK(X2, X376) in ABC.

K1262 is a central cubic with center O, point of inflexion on the curve.

K1262 is a member of the pencil is generated by K758 and the decomposed cubic which is the union of the Euler line and the circumcircle. This pencil also contains K764, K1260, K1261.

Points on K1262 :

• vertices Q1, Q2, Q3 of the Thomson triangle.

• reflections R1, R2, R3 of Q1, Q2, Q3 in O.

• foci of (C), conic with center the midpoint X(10304) of X(3), X(3524), inscribed in Q1Q2Q3. F1 and F2 are the real foci. Recall that X(3524) is the centroid of the Thomson triangle.

• sixth common point (apart O and the in/excenters) of K1262 and the Stammler hyperbola (S), also on pK(X6, X10304). This point is the reflection X(46945) of X(5646) in O, SEARCH = 10.2656634619011, also on the lines {74,2930}, {155,548}. Recall that X(5646) is the Lemoine point of the Thomson triangle and this is the sixth common point (apart G, O, Q1, Q2, Q3) of K1262 and the Jerabek-Thomson hyperbola (JT).

More generally, a diagonal rectangular hyperbola (H) passing through the in/excenters meets K1262 again at two points that lie on a line (L) passing through X(5646). Recall that X(5646) lies on these lines {2,1350}, {40,392}, {64,631}, {110,5085}, {182,3167}, {354,612}, {511,5544}, {1201,2177}, {1351,3819}.

– with (L) = {2,1350}, we find that (H) is the Wallace hyperbola, passing through {1, 2, 20, 63, 147, 194, 487, 488, 616, 617, 627, 628, 1670, 1671, 1764, 2128, 2582, 2583, 2896, etc} and the two points are X(2), X(46944).

– with (L) = {40,392}, we find that (H) is the Jerabek hyperbola of the excentral triangle, passing through {1, 9, 40, 188, 191, 366, 1045, 1050, 1490, 2136, 2949, 2950, 2951, 3174, etc} and the two points are X(40), X(46947).

– with (L) = {3,5646}, we find that (H) is the Stammler hyperbola as above, passing through {1, 3, 6, 155, 159, 195, 399, 610, 1498, 1740, 2574, 2575, 2916, 2917, 2918, 2929, 2930, 2931, 2935, 2948, 3216, etc} and the two points are X(3), X(46945).