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K1237

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X(25), X(394), X(493), X(494), X(10132), X(10133)

vertices of the cevian triangle of X(25)

X(25)-Ceva conjugate of X(394), on the line through X(493), X(494)

Geometric properties :

Consider the pivotal cubic pK(Ω = p:q:r, P = u,v,w). The polar conic of X(6) is a circle (C) if and only if P = F(Ω) where F is the quadratic transformation given by :

F(p:q:r) = b^2 c^2 (a^2+b^2+c^2) p^2+c^2 (a^2 b^2+b^4-4 a^2 c^2-3 b^2 c^2+2 c^4) p q-b^2 (4 a^2 b^2-2 b^4-a^2 c^2+3 b^2 c^2-c^4) p r+2 a^4 (-2 a^2+b^2+c^2) q r : : .

Its inverse is given by :

F-1(u:v:w) = a^2 (a^2-2 b^2-2 c^2) (a^2-b^2-c^2) u^2+a^2 (2 a^4-3 a^2 b^2+b^4-4 a^2 c^2+b^2 c^2) u v+a^2 (2 a^4-4 a^2 b^2-3 a^2 c^2+b^2 c^2+c^4) u w+2 a^4 (2 a^2-b^2-c^2) v w : : .

For any such cubic, the parallels through X(6) to the asymptotes meet the curve at six finite points on a same circle (C1), in a sense the first Lemoine circle of the cubic.

The reflections in X(6) of these same asymptotes also meet the curve at six finite points on a same circle (C2), in a sense the second Lemoine circle of the cubic.

***

Examples of pairs {Ω = X(i), P = X(j)}for these {i,j} : {3,193}, {6,4}, {32,6}, {184,25}, {1992,598}, {2482,524}, {3167,2}, {8911,372}, {26920,371}.

The pairs {6,4}, {3,193} correspond to K006, K707 (a central cubic with center K hence a decomposed polar conic (C)).

The pairs {32,6}, {2482,524} correspond to pKs decomposed into three cevian lines.

At last, the pair {184,25} gives K1237.

***

K006 and K1237 generate a pencil which contains a third pK which turns out to be the union of the symmedians of ABC. Hence, K006 and K1237 meet at nine points, three on each symmedian.

The circles (C), (C1), (C2) are the same for both cubics.