Orthotransversal
In Euclidean geometry, the orthotransversal of a point is the line defined as follows.[1][2]
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. In 1933, Indian mathematician K. Satyanarayana called this line an "ortho-line".[3]
Existence of it can proved by various methods such as a pole and polar, the dual of Desargues' involution theorem , and the Newton line theorem.[4][5]
The tripole of the orthotransversal is called the orthocorrespondent of P,[6][7] And the transformation P → P⊥, the orthocorrespondent of P is called the orthocorrespondence.[8]
Example
- The orthotransversal of the Feuerbach point is the OI line.[9][10]
- The orthotransversal of the Jerabek center is the Euler line.
- Orthocorrespondents of Fermat points are themselves.[11]
- 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.[1]
- The orthotransversal of a point on the circumcircle of the reference triangle ABC passes through the circumcenter of ABC.[1] Furthermore, the Steiner line, the orthotransversal, and the trilinear polar are concurrent.[12]
- The orthotransversals of a point P on the Euler line is perpendicular to the line through the isogonal conjugate and the anticomplement of P.[13]
- The orthotransversal of the nine-point center is perpendicular to the Euler line of the tangential triangle.[14]
- For the quadrangle ABCD, 4 orthotransversals for each component triangles and each remaining vertexes are concurrent.[15]
- Barycentric coordinates of the orthocorrespondent of P(p: q: r) are
where SA,SB,SC 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).[16] 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.[17]
- The orthopivotal cubic of the vertex is the isogonal image of the Apollonius circle (the Apollonian strophoid[18]).
See also
- Orthocenter
- Orthopole
- Orthologic triangles
- Transversal
Notes
- ↑ 1.0 1.1 1.2 Gibert, Bernard (2003). "Orthocorrespondence and Orthopivotal Cubics". Forum Geometricorum 3. http://www.bernard-gibert.fr/files/Resources/volume3a.pdf.
- ↑ Eliud Lozada, César. "Extended glossary". https://faculty.evansville.edu/ck6/encyclopedia/ext_glossary.html.
- ↑ Satyanarayana, K. (1933). "On Conics Having a Common Self-conjugate Triangle". The Mathematics Student 5: 19. https://dspace.bcu-iasi.ro/handle/123456789/44183.
- ↑ Cohl, Telv. "Extension of orthotransversal". https://artofproblemsolving.com/community/q1h507541p2851090.
- ↑ "Existence of Orthotransversal". https://artofproblemsolving.com/community/q1h2745598p23957981.
- ↑ Bernard, Gibert (2003). "Antiorthocorrespondents of Circumconics". Forum Geometricorum 3.
- ↑ Gibert, Bernard; van Lamoen, Floor (2003). "The Parasix Configuration and Orthocorrespondence". Forum Geometricorum 3: 173.
- ↑ Evers, Manfred (2012). "Generalizing Orthocorrespondence". Forum Geometricorum 12.
- ↑ Li4; S⊗; 和輝. "幾何引理維基" (in zh). https://lii4.github.io/Geometry_Lemma_Wiki.pdf.
- ↑ 10.0 10.1 Mathworld Orthocorrespondent.
- ↑ dagezjm. "Pedal triangle". https://artofproblemsolving.com/community/q1h2175301p17775874.
- ↑ Li4. "圓錐曲線" (in zh). https://lii4.github.io/Conic.pdf.
- ↑ Li4; S. "張志煥截線" (in zh). https://permutation-chang.github.io/Permutationline.pdf.
- ↑ S. "正交截線" (in zh). https://permutation-chang.github.io/Orthotransversal.pdf.
- ↑ "QA-Tf14: QA-Orthotransversal Point". https://chrisvantienhoven.nl/qa-items/qa-transformations/qa-tf14.
- ↑ "Orthopivotal Cubics". http://www.bernard-gibert.fr/gloss/orthopivotalcubi.html.
- ↑ Gibert, Bernard. "Neuberg Cubics". http://bernard-gibert.fr/files/Resources/neubergs.pdf.
- ↑ "K053". http://bernard-gibert.fr/Exemples/k053.html.
References
- Pohoata, Cosmin; Zajic, Vladimir (2008). "Generalization of the Apollonius Circles". arXiv:0807.1131 [math.HO].
- Evers, Manfred (2019). "On the Geometry of a Triangle in the Elliptic and in the Extended Hyperbolic Plane". arXiv:1908.11134 [math.MG].
External links
- Weisstein, Eric W.. "Orthotransversal". http://mathworld.wolfram.com/Orthotransversal.html.
- Weisstein, Eric W.. "Orthocorrespondent". http://mathworld.wolfram.com/Orthocorrespondent.html.
- Li4. "平面幾何" (in zh). https://lii4.github.io/Plane_Geometry.pdf.
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