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X(5), X(11), X(12), X(523), X(1312), X(1313), X(8819), X(8901), X(8902)

A'B'C' : cevian triangle of X(5)

PaPbPc : anticevian triangle of X(523)

Fa, Fb, Fc : vertices of the Feuerbach triangle, extraversions of X(11)

Qa, Qb, Qc : X(115)-isoconjugates of Fa, Fb, Fc and extraversions of X(12)

Ra, Rb, Rc : X(5)-Ceva conjugates of of Fa, Fb, Fc and extraversions of X(8819)

infinite points of the rectangular circum-hyperbola passing through X(140)

K672 is the locus of M such that the anticevian triangle of M and the Feuerbach triangle are perspective at P which lies on K877.

K877 = pK(X115, X5) has three real asymptotes :

• one is the line X(5)X(523), the perpendicular bisector of OH,

• two are the parallels at X(10277) to the asymptotes of the rectangular circum-hyperbola passing through X(140).

The orthic line is the line X(5)X(399) and the polar conic of X(5) is a diagonal rectangular hyperbola (H) passing through X(1), X(5), X(30), X(395), X(396), X(523), X(1749), the excenters and Pa, Pb, Pc. Its center is X(476). Hence, the tangents at Pa, Pb, Pc, X(523) concur at X(5).

The polar conic of X(523) passes through X(11), Fa, Fb, Fc hence the tangents at these points are parallel and perpendicular to the Euler line.

K877 contains the vertices of six triangles which are all two by two perspective at a point which also lies on the cubic. These are the triangles mentioned above with A-vertices A, A', Pa, Fa, Qa, Ra. The tangents at the vertices of each triangle concur on the cubic. It follows that K877 is a pK in any of these triangles and, actually, in infinitely many other triangles. For example, K877 is the pK with pivot X(523), isopivot X(5) in PaPbPc.

Collinearities on K877 :

X5 X11 X12

X5 X1312 X1313

X11 X8819 X8901

X12 X523 X8819

X523 X8901 X8902

Tangentials on K877 :

X8901 is the tangential of X5

which is the tangential of X523

which is the tangential of X11

X8902 is the tangential of X12