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X(1), X(2), X(3), X(15), X(16), X(17), X(18), X(39), X(2896), X(3413), X(3414)

midpoints of ABC


points of K655 on (O)


Q140 is a circular quartic passing through the X(5)-anticevian points. See Table 28.

There are 10 (not always all real) such points. They are represented in black and green on the figure.

For example, the nine-point center of the anticevian triangle of N' is N' itself. O' and H' are the corresponding circumcenter and orthocenter.


The singular focus of Q104 is X(2080).

Q140 has two real asymptotes parallel to those of the Kiepert hyperbola at X =

2a^4+b^4+c^4+4 b^2 c^2-4 a^2 b^2-4 a^2 c^2 : : ,

with SEARCH = 8.11132214901875.

X lies on the lines {99,754}, {194,6680}, {230,538}, {325,543}, etc.

X is now X(14148) in ETC (2017-08-21).

The remaining points on the sidelines of ABC are

{0 : -a^2+2 b^2+c^2 : a^2-b^2-2 c^2}, {2 a^2-b^2+c^2 : 0 : -a^2+b^2-2 c^2}, {2 a^2+b^2-c^2 : -a^2-2 b^2+c^2 : 0}.