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X(3), X(54), X(2574), X(2575), X(6759), X(13557)

vertices of the CircumNormal triangle

four X3-OAP points, see Table 53

X(5)-circumcevian points (blue points)

foci of the inconic with center X(140)

other points below

Q141 is a circular quartic with singular focus X(186). It meets the line at infinity again at the same points as the Jerabek hyperbola. The tangents at these points meet at X on the lines {3,1568}, {113,2071}, {125,539}, {140,389}, etc, with SEARCH = 3.83823890657063. X is now X(14156) in ETC (2017-08-26).

Q141 passes through X(3), a point of inflexion with tangent passing through X(49). The three remaining intersections E1, E2, E3 with the Euler line lie on the cubic pK(X6, Z) with Z on the lines {4,54}, {24,1192}, {30,110}, {74,186}, {112,1971}, {113,3153}, {154,378}, {156,382}, etc, with SEARCH = -5.64972330061297. Z is now X(14157) in ETC (2017-08-26).

The tangents at A, B, C pass through X(3).

Q141 meets the Napoleon cubic K005 at A, B, C, X(3), X(54) and seven other points which are the X(5)-circumcevian points. These points also lie on Q024 and Q037.


The X(5)-circumcevian points

Consider a point P with circumcevian triangle A'B'C'.

Its circumcenter is obviously O = X(3) and let us denote by Q, N its orthocenter and its nine-point center respectively.

P is said to be a X(5)-circumcevian point if P and N coincide i.e. P is the nine-point center of its circumcevian triangle.

There are seven such points but they are not necessarily all real. In the opposite figure, five only are real.


Recall that the locus of P such that P, N, X(3) [resp. X(5)] are collinear is Q037 together with (O) [resp. Q024 together with the line at infinity].

More generally, the locus of P such that P, N and a fixed point M are collinear is a tricircular circum-septic passing through M and the X(5)-circumcevian points.