Orthotransversal

In Euclidean geometry, the orthotransversal of a point is the line defined as follows.

For a triangle and a point , three orthotraces, intersections of lines and perpendiculars of through respectively are collinear. The line which includes these three points is called the orthotransversal of . In 1933, Indian mathematician K. Satyanarayana called this line an "ortho-line".

Existence of it can proved by various methods such as a pole and polar, the dual of , and the Newton line theorem.

The tripole of the orthotransversal is called the orthocorrespondent of , And the transformation → , the orthocorrespondent of is called the orthocorrespondence.

Example

• The orthotransversal of the Feuerbach point is the OI line.
• The orthotransversal of the Jerabek center is the Euler line.
• Orthocorrespondents of Fermat points are themselves.
• The orthocorrespondent of the Kiepert center X(115) is the focus of the Kiepert parabola X(110).

Properties

• There are exactly two points which share the orthoccorespondent. This pair is called the antiorthocorrespondents.
• The orthotransversal of a point on the circumcircle of the reference triangle passes through the circumcenter of . Furthermore, the Steiner line, the orthotransversal, and the trilinear polar are concurrent.
• The orthotransversals of a point P on the Euler line is perpendicular to the line through the isogonal conjugate and the anticomplement of P.
• The orthotransversal of the nine-point center is perpendicular to the Euler line of the tangential triangle.
• For the quadrangle , 4 orthotransversals for each component triangles and each remaining vertexes are concurrent.
• Barycentric coordinates of the orthocorrespondent of are

where are Conway notation.

Orthopivotal cubic

The Locus of points that , and are collinear is a cubic curve. This is called the orthopivotal cubic of , . Every orthopivotal cubic passes through two Fermat points.

Example

• is the line at infinity and the Kiepert hyperbola.
• is the Neuberg cubic.
• The orthopivotal cubic of the vertex is the isogonal image of the Apollonius circle (the Apollonian strophoid).

See also

• Orthocenter
• Orthopole
• Orthologic triangles
• Transversal

Notes

References

External links