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K701

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X(2), X(22), X(25)

M1, M2 see below

K701 has the remarkable property to be an isogonal pK in three different triangles which are all real when ABC is acutangle.

The corresponding pivots are X(25) and the intersections M1, M2 of the line (L) passing through X(6), X(25) and the rectangular circum-hyperbola (H) passing through X(1370).

Indeed, (L) is the orthic line of K701 but also the orthic line of the Hessian H701 of K701. It follows that the polar conics of each of the three points X(25), M1, M2 in both cubics are rectangular hyperbolas intersecting at four points which are the in/excenters of a corresponding triangle in which K701 is an isogonal pK. These three triangles are represented in the figures below, the in/excenters being the red points and the pivot being the blue point. The polar conics in K701, H701 are the orange and cyan curves respectively. The shape of ABC is always the same but not its size.

More informations in the paper Bi-isogonal and Tri-isogonal Pivotal Cubics. See also the analogous cubic K719.

K701a K701b
K701c