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

X(1), X(145), X(2136), X(2137), X(3445)

excenters

vertices of the cevian triangle of X(145)

other points below

Geometric properties :

We meet K1086 in quite a few different contexts where further properties are given :

• How pivotal cubics intersect the circumcircle, see figure 7 p.218.

• Cubics Related to Coaxial Circles, see figure 11 p.90.

• K199, K830, Q104 concerning its points at infinity. In particular, for Ω on psK(X55, X2, X6) and P on the complement of K259 = psK(X55, X2, X1), one can find a pK(Ω, P) sharing its points at infinity with K1086. Other examples : pK(X7, X6644), pK(X268, X280), pK(X281, X4).

• K360, K632, K921 concerning its points on the circumcircle Q1, Q2, Q3. In particular, for Ω on psK(X32 x X56, X56, X6) and P on K360 = psK(X56, X7, X1), one can find a pK(Ω, P) sharing its points on the circumcircle with K1086. Other examples : pK(X2175, X218), pK(X3209, X4).

Notes :

• Q1, Q2, Q3 lie on the homothetic of the polar conic of X(145) under h(X145, 1/2). This polar conic is a rectangular hyperbola which contains X(1), X(145), the vertices of the anticevian triangle of X(145). It is tangent at X(145) to the line X(145)X(3445).

• The triangle Q1 Q2 Q3 is a Poncelet poristic triangle with orthocenter X(145).

• (C) is the bicevian conic C(X2, X145), bitangent to the incircle at its intersections with the line X(1)X(2). It contains the third points R1, R2, R3 of K1086 on the sidelines of Q1 Q2 Q3. These are the vertices of the pedal triangle of X(3445) in triangle Q1 Q2 Q3.

• The Simson line of each point Qi is tangent to the incircle and parallel to the opposite sideline of Q1 Q2 Q3. These three Simson lines bound a triangle S1 S2 S3 which is the reflection of Q1 Q2 Q3 in X(1) hence with orthocenter X(8).

• Since X(145) is the anticomplement of X(8), the Simson lines of the antipodes of Q1 Q2 Q3 on the circumcircle concur at X(8).