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.
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.[3][4]
The tripole of the orthotransversal is called the orthocorrespondent of P,[5][6] And the transformation P → P⊥, the orthocorrespondent of P is called the orthocorrespondence.[7]
Example
[edit]- 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).
Properties
[edit]- 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 ABC passes through the circumcenter of ABC.[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 ABCD, 4 orthotransversals for each component triangles and each remaining vertexes are concurrent.[14]
- Barycentric coordinates of the orthocorrespondent of P(p: q: r) are
where SA,SB,SC are Conway notation.
Orthopivotal cubic
[edit]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).[15] Every orthopivotal cubic passes through two Fermat points.
Example
[edit]- O(X2) is the line at infinity and the Kiepert hyperbola.
- O(X3) 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
[edit]Notes
[edit]- ^ a b c Gibert, Bernard (2003). "Orthocorrespondence and Orthopivotal Cubics" (PDF). Forum Geometricorum. 3.
- ^ Eliud Lozada, César. "Extended glossary". faculty.evansville.edu.
- ^ Cohl, Telv. "Extension of orthotransversal". AoPS.
- ^ "Existence of Orthotransversal". AoPS.
- ^ 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" (PDF) (in Chinese).
{{cite web}}: CS1 maint: numeric names: authors list (link) - ^ a b Mathworld Orthocorrespondent.
- ^ dagezjm. "Pedal triangle". AoPS.
- ^ Li4. "圓錐曲線" (PDF) (in Chinese).
{{cite web}}: CS1 maint: numeric names: authors list (link) - ^ Li4; S. "張志煥截線" (PDF) (in Chinese).
{{cite web}}: CS1 maint: numeric names: authors list (link) - ^ S. "正交截線" (PDF) (in Chinese).
- ^ "QA-Tf14: QA-Orthotransversal Point". ENCYCLOPEDIA OF QUADRI-FIGURES (EQF). Retrieved 2024-11-02.
- ^ "Orthopivotal Cubics". Catalogue of Triangle Cubics.
- ^ Gibert, Bernard. "Neuberg Cubics" (PDF).
- ^ "K053". Cubic in Triangle Plane.
References
[edit]- 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
[edit]- 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)