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X(3), X(6), X(574)

vertices of these triangles :

• Thomson (points Qi)

• tangential of Thomson (points Ri)

other points below

Geometric properties :

K1125 is a member of the pencil of cubics generated by K172 and K1114. This is studied in Table 71.

X(574), on the Brocard axis, is a point of inflexion whose harmonic polar is the Lemoine axis.

Other noticeable points on K1125 :

P1 = a^2 (2 a^6 b^2-2 a^2 b^6+2 a^6 c^2+a^4 b^2 c^2+4 a^2 b^4 c^2-3 b^6 c^2+4 a^2 b^2 c^4+6 b^4 c^4-2 a^2 c^6-3 b^2 c^6) : : , SEARCH = -0.291928054048584, on lines {2,3}, {51,9465}, {111,263}, {154,20965}, {353,1495}, etc.

P2 = a^2 (10 a^6 b^4-12 a^4 b^6+2 a^2 b^8+19 a^6 b^2 c^2-9 a^4 b^4 c^2-11 a^2 b^6 c^2+b^8 c^2+10 a^6 c^4-9 a^4 b^2 c^4-6 a^2 b^4 c^4-b^6 c^4-12 a^4 c^6-11 a^2 b^2 c^6-b^4 c^6+2 a^2 c^8+b^2 c^8) : : , SEARCH = -2.67508701753819, on lines {2,9743}, {6,160}, {110,8722}, etc, and JT.

P3 = a^2 (20 a^12 b^4-96 a^10 b^6+104 a^8 b^8-28 a^4 b^12+41 a^12 b^2 c^2-255 a^10 b^4 c^2+106 a^8 b^6 c^2+130 a^6 b^8 c^2+45 a^4 b^10 c^2-67 a^2 b^12 c^2+20 a^12 c^4-255 a^10 b^2 c^4+12 a^8 b^4 c^4+86 a^6 b^6 c^4+48 a^4 b^8 c^4+129 a^2 b^10 c^4-40 b^12 c^4-96 a^10 c^6+106 a^8 b^2 c^6+86 a^6 b^4 c^6-130 a^4 b^6 c^6-62 a^2 b^8 c^6+96 b^10 c^6+104 a^8 c^8+130 a^6 b^2 c^8+48 a^4 b^4 c^8-62 a^2 b^6 c^8-112 b^8 c^8+45 a^4 b^2 c^10+129 a^2 b^4 c^10+96 b^6 c^10-28 a^4 c^12-67 a^2 b^2 c^12-40 b^4 c^12) : : , SEARCH = 0.579517524972630, on line {1350,26714} and K1124.

These points are P1 = X(34098), P2 = X(34099) in ETC (2019-08-21).