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X(1), X(2), X(6), X(20), X(25), X(64), X(69), X(159), X(200), X(269), X(1763), X(2138), X(2139), X(7097), X(13575), X(17742), X(40187), X(40188), X(40189), X(40190), X(40219), X(40220), X(40221), X(40222), X(40223), X(40224), X(40225), X(40226), X(40227)


cevians of X(69)

K169 is the isogonal pK with pivot the isotomic conjugate X(69) of the orthocenter. It is a member of the Thomson-Grebe pencil. See Table 13.

Its asymptotes are parallel to the Simson lines that pass through K.

K169 is the isotomic transform of pK(X76, X305).

It meets the line at infinity and the Steiner ellipse at the same points as pK(X2, X315) and pK(X2, X76) = K141 respectively.

The symbolic substitution SS{a -> √a} transforms K169 into K308. See also the associated K1162.


Points on the circum-circle (O)

K169 meets (O) at A, B, C (with tangents passing through the isopivot X25) and three other points T1, T2, T3 which also lie on a good number of other cubics, in particular a family of pKs with pole Ω on K177, pivot P on K141 = gK177, isopivot Q on K174.

K177 = pK(X32, X2) through 2, 3, 6, 25, 32, 66, 206, 1676, 1677, 3162, 19615, 41378, 41379, 52041.

K141 = pK(X2, X76) through 2, 4, 6, 22, 69, 76, 1670, 1671, 18018, 19613, 41361.

K174 = pK(X32, X22) through 3, 6, 22, 25, 159, 2353, 20993, 34207, 34427.

Examples :

pK(X2, X76) = K141, pK(X3, X2) = K168, pK(X6, X69) = K169, pK(X25, X41361), pK(X32, X22) = K174, pK(X66, gX19615), pK(X206, X6), pK(X3162, X4), pK(X41378, X1670), pK(X41379, X1671).

Note that the inconic (of ABC) with center X(141), perspector X(76) is also inscribed in the triangle T1T2T3.

The isogonal conjugation with respect to T1T2T3 transforms K169 into the central cubic K1361.

See the analogous pairs {K002, K758}, {K003, K1267}, {K006, K1362}, {K007, K1327}. The same process keeps K004 unchanged.