Orthotransversal Explained

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

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 .

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

The tripole of the orthotransversal is called the orthocorrespondent of,^{[5] [6]} And the transformation →, the orthocorrespondent of is called the orthocorrespondence.^[7]

Example

• The orthotransversal of the Feuerbach point is the OI line.^[8]
• The orthotransversal of the Jerabek center is the Euler line.
• Orthocorrespondents of Fermat points are themselves.^[9]
• 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.^[10] 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.^[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, 4 orthotransversals for each component triangles and each remaining vertexes are concurrent.^[14]
• Barycentric coordinates of the orthocorrespondent of are

p( - pS _{A+qS} _{B+rS} ^{2 qr:q(pS} ^{A - qS} _{B+rS} ^{2 rp:r(pS} ^{A + qS} _{B-rS} ^{2 pq,} ^{C) + c}

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, .^[15] Every orthopivotal cubic passes through two Fermat points.

Example

• is the line at infinity and the Kiepert hyperbola.
• is the Neuberg cubic.^[16]
• The orthopivotal cubic of the vertex is the isogonal image of the Apollonius circle (the Apollonian strophoid^[17]).

See also

• Orthocenter
• Orthopole
• Orthologic triangles
• Transversal

References

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

External links

• • • Web site: Li4 . 平面幾何 . zh.

Notes and References

1. Gibert . Bernard . 2003 . Orthocorrespondence and Orthopivotal Cubics . . 3.
2. Web site: Eliud Lozada . César . Extended glossary . faculty.evansville.edu.
3. Web site: Cohl . Telv . Extension of orthotransversal . AoPS.
4. Web site: Existence of Orthotransversal . AoPS.
5. Bernard . Gibert . 2003 . Antiorthocorrespondents of Circumconics . Forum Geometricorum . 3.
6. Gibert . Bernard . van Lamoen . Floor . 2003 . The Parasix Configuration and Orthocorrespondence . Forum Geometricorum . 3 . 173.
7. Evers . Manfred . 2012 . Generalizing Orthocorrespondence . Forum Geometricorum . 12.
8. Web site: Li4 . S⊗ . 和輝 . 幾何引理維基 . zh.
9. Web site: dagezjm . Pedal triangle . AoPS.
10. [MathWorld|Mathworld]
11. Web site: Li4 . 圓錐曲線 . zh.
12. Web site: Li4 . S . 張志煥截線 . zh.
13. Web site: S . 正交截線 . zh.
14. Web site: QA-Tf14: QA-Orthotransversal Point . 2024-11-02 . ENCYCLOPEDIA OF QUADRI-FIGURES (EQF).
15. Web site: Orthopivotal Cubics . Catalogue of Triangle Cubics.
16. Web site: Gibert . Bernard . Neuberg Cubics .
17. Web site: K053 . Cubic in Triangle Plane.