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K719

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X(5), X(252)

P1, P2, P3 described below

vertices of the cevian triangle of X(5)

K719 has the remarkable property to be an isogonal pK with respect to three different triangles represented in the figures below.

In other words, K719 is a tri-isogonal pK. See Bi-isogonal and Tri-isogonal Pivotal Cubics. See also the analogous cubic K701.

The corresponding pivots P1, P2, P3 are its intersections with its orthic line, namely the Napoleon line X(6), X(17), X(18), etc.

K719P1 K719P2
K719P3

Each point Pi is associated with a (yellow) triangle Ti such that K719 is the isogonal pK with pivot Pi with respect to Ti.

The dashed green conics are the polar conics of each point Pi with respect to K719 and its Hessian cubic. They intersect at the in/excenters of each triangle Ti (dark green points).

Denote by Qi the tangential of each point Pi with respect to K719.

The blue conics are the polar conics of Qi with respect to K719 and its Hessian cubic. They intersect at Pi and the vertices of the corresponding triangle Ti.