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X(3), X(4), X(3357), X(59290), X(59291), X(59292), X(59293)

X(59360) → X(59371)

infinite points of the altitudes, on K004

vertices of the CircumTangential triangle T1T2T3, on K024

points Q1, Q2, Q3 of K006 on (O)

Ha, Hb, Hc on the altitudes and on the perpendiculars at O to the cevian lines of O.

Oa, Ob, Oc on the cevian lines of O and on the lines passing through H and the reflections of the vertices of ABC in its perpendicular bisectors. The centroid of this triangle OaObOc is G.

These triangles HaHbHc and OaObOc are perspective at the midpoint X(3357) of X(3), X(64) which is a point on K1347.

Geometric properties :

Let spK(P, X5) be a MacBeath cubic as in Table 80 meeting the sidelines of ABC at A', B', C'.

A'B'C' is the pedal triangle of some point Q if and only if P lies on K617, and, in this case, Q lies on K1347.

For example, with P = X(4), X(20), X(68), the corresponding point Q is X(3), X(3357), X(4) and the cubics are K009, K846, K1318 respectively.

K1347 is a nodal cubic with node O and nodal tangents parallel to the asymptotes of the Jerabek hyperbola.

The isogonal conjugation with respect to Q1Q2Q3 transforms K1347 into K028. Recall that ABC and Q1Q2Q3 both circumscribe the MacBeath inconic with foci O and H.

***

A construction

Let P1, P2 be two antipodes on (O) and let (L1), (L2) be the parallels at O to the Steiner lines of P1, P2 respectively.

(L1), (L2) meet the lines (HP1), (HP2) at Q1, Q2 respectively, which are two points on K1347 and on a line passing through X(3357).

***

Further properties of OaObOc

OaObOc is perspective to

• the cevian triangle of P for P on a pK passing through {2, 3, 69, 317, 2042}.

• the anticevian triangle of P for P on a pK passing through {2, 3, 68,2052}.

In both cases, the perspector Q lies on a pK passing through {3, 4, 1217}.

• the circumcevian triangle of P for P on a circum-quartic passing through {3, 24, 523}.

• the pedal triangle of P for P on a cubic passing through {3, 20}.

• the antipedal triangle of P for P on a circum-cubic passing through {4, 64}.

Example : the cevian triangle of X(69) = pedal triangle of X(20) corresponds to a perspector on the lines {X2, X578}, {X3, X68}, {X4, X69}, {X5, X394}, {X20, X2888}, {X66, X1350}, SEARCH = 19.4116951150284, lying on the cubic above.

OaObOc is orthologic to

• the cevian triangle of P for P on K617.

• the anticevian triangle of P for P on K260.

In both cases, the orthology centers lie on the same cubics passing through {3, 4, 3357} and {25, 1370, 1660}.

• the circumcevian triangle of P for P on the van Aubel Line passing through {4, 6, 53, 217, 387, 393, 397, 398, 1172, 1181, 1199, 1249, 1498, 1503, 1514, 1515, 1540, 1547, 1548, 1549, 1587, 1588, 1834, 1865, 1901, 1990, 2207, 2211, 2442, 2883, 3070, 3071, 3087, 3092, 3093, 3194, etc}. In this case, the orthology centers lie on a hyperbola passing through {2, 3} and on the trilinear polar of X(112) passing through {6, 25, 51, 154, 159, 161, 184, 206, 232, 1194, 1474, 1495, 1660, 1843, 1915, 1971, 1974, 2194, 2203, 2299, 2393, 2445, 3192, etc}.