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∑ [b^2c^2 x(y + z - 2x) + 3a^4 y z](y -z) = 0

X(2), X(6), X(111), X(524), X(9217), X(52695), X(52696), X(52697)

see points below

Geometric properties :

K1306 is a circular cubic with singular focus F = X(38698), the member K(1/3) of the pencil mentioned in K1156. See also K1307.

OF = 1/3 OX(111), (vectors).

Locus property

Let P be a point and (La), (Lb), (Lc) the polar lines of A, B, C in the circles PBC, PCA, PAB respectively.

(La), (Lb), (Lc) determine a triangle (T) with vertices A1, B1, C1 where A1 = (Lb) ∩ (Lc), etc.

(T) is perspective to the medial triangle MaMbMc when P lies on K1306.

Other points on K1306

The line GK meets the sidelines of ABC at U, V, W. The cubic K1306 meets these sidelines at A', B', C' such that A' is the homothetic of U under h(Ma, 1/3), B' and C' cyclically.

P1 = X(52695) = 7 a^4-7 a^2 b^2+b^4-7 a^2 c^2+5 b^2 c^2+c^4 : : , SEARCH = 3.16894256659498, on the lines {2,111}, {147,376}, {524,9217}.

P2 = X(52697) = a^2 (3 a^6-a^4 b^2-3 a^2 b^4+b^6-a^4 c^2+a^2 b^2 c^2+b^4 c^2-3 a^2 c^4+b^2 c^4+c^6) : : , SEARCH = 0.936011481089684, on the lines {6,111}, {182,399}.

P3 = X(52696) = a^2 (a^2+b^2) (a^2+c^2) (a^4-3 a^2 b^2+b^4-3 a^2 c^2+3 b^2 c^2+c^4) : : , SEARCH = -5.09506346766827, on the lines {2,32}, {187,827}, {P2,9217}, the tangential of G.

Remarks (see Ivan Pavlov, Euclid #5571) :

• (T) is perspective to ABC when P lies on K018, together with (L∞) where the perspector is G, (O) where the perspector is K, and a tricircular sextic obtained when (La), (Lb), (Lc) are concurrent.

• (T) is orthologic to ABC when P lies on K001, together with the same curves above. One of the centers of orthology lies on K005.