EulerConics

Euler conics

There are two conics related to the Euler line of triangle ABC through the trilinear polar/pole construction. The conics are dual to each other (see TriangleConics.html ). The circumconic is a hyperbola and the dual inconic is a parabola. Their perspector is the tripolar D of the Euler line with respect to ABC. A*B*C* is the triangle with vertices the harmonic associates of D with respect to ABC. The trilinear polars (tripolars) of points P on the Euler line with respect to A*B*C* envelope the hyperbola and the trilinear polars of these points with respect to ABC envelope the parabola, therefore the name I use.
The parabola has its focus E at X(112), which is on the circumcircle. Its tangent at the vertex is the Wallace-Simson line of E and passes from the symmedian point K of the triangle. Thus, this conic is easy to construct by locating E through its corresponding Steiner line, which is the line through {H,K} (see SteinerReflected.html ).
The picture shows also the axes and focal points F₁, F₂ of the hyperbola.

Euler conics

See Also