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# Orthic Inconic

The orthic inconic of a triangle is the inconic with inconic parameters

 (1)

It has trilinear equation

 (2)

(P. Moses, pers. comm., Feb. 8, 2005), where , , and are Conway triangle notation.

It is an ellipse for acute triangles and a hyperbola for obtuse triangles.

When the orthic inconic is an inellipse, it has area

 (3)

where is the area of the reference triangle.

It has the orthocenter as its Brianchon point and the symmedian point as its center. Its contact points with the reference triangle form the orthic triangle, which is also its polar triangle.

The orthic inconic passes through Kimberling centers for (the center of the Jerabek hyperbola) and 2969.

The axes of the orthic inconic are parallel to the asymptotes of the Jerabek hyperbola.

Brianchon Point, Inconic, Inellipse, MacBeath Circumconic, Orthic Triangle

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## References

Parry, C. F. "The Isogonal Tripolar Conic." Forum Geom. 1, 33-42, 2001. http://forumgeom.fau.edu/FG2001volume1/FG200106index.html.

Orthic Inconic

## Cite this as:

Weisstein, Eric W. "Orthic Inconic." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/OrthicInconic.html