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

For a triangle ABC and a point P, three orthotraces, intersections of lines BC, CA, AB and perpendiculars of AP, BP, CP through P respectively are collinear. The line which includes these three points is called the orthotransversal of P.

Existence of it can proved by various methods such as a pole and polar, the dual of Desargues' involution theorem [ru] , and the Newton line theorem.

The tripole of the orthotransversal is called the orthocorrespondent of P, And the transformation P → P, the orthocorrespondent of P 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 ABC passes through the circumcenter of ABC. 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 ABCD, 4 orthotransversals for each component triangles and each remaining vertexes are concurrent.
• Barycentric coordinates of the orthocorrespondent of P(p: q: r) are

${\displaystyle p(-pS_{A}+qS_{B}+rS_{C})+a^{2}qr:q(pS_{A}-qS_{B}+rS_{C})+b^{2}rp:r(pS_{A}+qS_{B}-rS_{C})+c^{2}pq,}$

where S_A,S_B,S_C are Conway notation.

Orthopivotal cubic

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

Example

• O(X₂) is the line at infinity and the Kiepert hyperbola.
• O(X₃) 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

1. ^ Gibert, Bernard (2003). "Orthocorrespondence and Orthopivotal Cubics" (PDF). Forum Geometricorum. 3.
2. Eliud Lozada, César. "Extended glossary". faculty.evansville.edu.
3. Cohl, Telv. "Extension of orthotransversal". AoPS.
4. "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. Li4; S⊗; 和輝. "幾何引理維基" (PDF) (in Chinese).{{cite web}}: CS1 maint: numeric names: authors list (link)
9. ^ Mathworld Orthocorrespondent.
10. dagezjm. "Pedal triangle". AoPS.
11. Li4. "圓錐曲線" (PDF) (in Chinese).{{cite web}}: CS1 maint: numeric names: authors list (link)
12. Li4; S. "張志煥截線" (PDF) (in Chinese).{{cite web}}: CS1 maint: numeric names: authors list (link)
13. S. "正交截線" (PDF) (in Chinese).
14. "QA-Tf14: QA-Orthotransversal Point". ENCYCLOPEDIA OF QUADRI-FIGURES (EQF). Retrieved 2024-11-02.
15. "Orthopivotal Cubics". Catalogue of Triangle Cubics.
16. Gibert, Bernard. "Neuberg Cubics" (PDF).
17. "K053". Cubic in Triangle Plane.

References

• Cosmin Pohoata, Vladimir Zajic (2008). "Generalization of the Apollonius Circles". arXiv:0807.1131.
• Manfred Evers (2019), "On The Geometry of a Triangle in the Elliptic and in the Extended Hyperbolic Plane". arXiv:1908.11134

External links

• Weisstein, Eric W. "Orthotransversal". MathWorld.
• Weisstein, Eric W. "Orthocorrespondent". MathWorld.
• Li4. "平面幾何" (PDF) (in Chinese).{{cite web}}: CS1 maint: numeric names: authors list (link)

• Triangle geometry
• Straight lines defined for a triangle