Orthotransversal

In Euclidean geometry, the orthotransversal of a point is the line defined as follows.^[1]^[2]

For a triangle Template:Mvar and a point Template:Mvar, three orthotraces, intersections of lines Template:Mvar and perpendiculars of Template:Mvar through Template:Mvar respectively are collinear. The line which includes these three points is called the orthotransversal of Template:Mvar.

Existence of it can proved by various methods such as a pole and polar, the dual of Template:Interlanguage link , and the Newton line theorem.^[3]^[4]

The tripole of the orthotransversal is called the orthocorrespondent of Template:Mvar,^[5]^[6] And the transformation Template:Mvar → Template:Math, the orthocorrespondent of Template:Mvar is called the orthocorrespondence.^[7]

Example

The orthotransversal of the Feuerbach point is the OI line.^[8]^[9]
The orthotransversal of the Jerabek center is the Euler line.
Orthocorrespondents of Fermat points are themselves.^[10]
The orthocorrespondent of the Kiepert center X(115) is the focus of the Kiepert parabola X(110).

There are exactly two points which share the orthoccorespondent.^[9] This pair is called the antiorthocorrespondents.^[1]
The orthotransversal of a point on the circumcircle of the reference triangle Template:Mvar passes through the circumcenter of Template:Mvar.^[1] Furthermore, the Steiner line, the orthotransversal, and the trilinear polar are concurrent.^[11]
The orthotransversals of a point P on the Euler line is perpendicular to the line through the isogonal conjugate and the anticomplement of P.^[12]
The orthotransversal of the nine-point center is perpendicular to the Euler line of the tangential triangle.^[13]
For the quadrangle Template:Mvar, 4 orthotransversals for each component triangles and each remaining vertexes are concurrent.^[14]
Barycentric coordinates of the orthocorrespondent of Template:Math are

$p (- p S_{A} + q S_{B} + r S_{C}) + a^{2} q r : q (p S_{A} - q S_{B} + r S_{C}) + b^{2} r p : r (p S_{A} + q S_{B} - r S_{C}) + c^{2} p q,$

The Locus of points Template:Mvar that Template:Math, and Template:Mvar are collinear is a cubic curve. This is called the orthopivotal cubic of Template:Mvar, Template:Math.^[15] Every orthopivotal cubic passes through two Fermat points.

Cosmin Pohoata, Vladimir Zajic (2008). "Generalization of the Apollonius Circles". Template:Arxiv.
Manfred Evers (2019), "On The Geometry of a Triangle in the Elliptic and in the Extended Hyperbolic Plane". Template:Arxiv