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 [ru] , 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

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

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 $Q$ , $O(Q)$ .^[16] Every orthopivotal cubic passes through two Fermat points.

Example

$O(X 2)$ is the line at infinity and the Kiepert hyperbola.
$O(X 3)$ is the Neuberg cubic.^[17]
The orthopivotal cubic of the vertex is the isogonal image of the Apollonius circle (the Apollonian strophoid^[18]).

Notes

^ ^a ^b ^c Gibert, Bernard (2003). "Orthocorrespondence and Orthopivotal Cubics" (PDF). Forum Geometricorum. 3.
^ Eliud Lozada, César. "Extended glossary". faculty.evansville.edu.
^ Satyanarayana, K. (1933). "On Conics Having a Common Self-conjugate Triangle". The Mathematics Student. 5: 19.
^ 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

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

[:0-1] Gibert, Bernard (2003). "Orthocorrespondence and Orthopivotal Cubics" (PDF). Forum Geometricorum. 3.

[2] Eliud Lozada, César. "Extended glossary". faculty.evansville.edu.

[3] Satyanarayana, K. (1933). "On Conics Having a Common Self-conjugate Triangle". The Mathematics Student. 5: 19.

[4] Cohl, Telv. "Extension of orthotransversal". AoPS.

[5] "Existence of Orthotransversal". AoPS.

[6] Bernard, Gibert (2003). "Antiorthocorrespondents of Circumconics". Forum Geometricorum. 3.

[7] Gibert, Bernard; van Lamoen, Floor (2003). "The Parasix Configuration and Orthocorrespondence". Forum Geometricorum. 3: 173.

[8] Evers, Manfred (2012). "Generalizing Orthocorrespondence". Forum Geometricorum. 12.

[9] Li4; S⊗; 和輝. "幾何引理維基" (PDF) (in Chinese).{{cite web}}: CS1 maint: numeric names: authors list (link)

[:1-10] Mathworld Orthocorrespondent.

[11] zjm. "Pedal triangle". AoPS.

[12] Li4. "圓錐曲線" (PDF) (in Chinese).{{cite web}}: CS1 maint: numeric names: authors list (link)

[13] Li4; S. "張志煥截線" (PDF) (in Chinese).{{cite web}}: CS1 maint: numeric names: authors list (link)

[14] S. "正交截線" (PDF) (in Chinese).

[15] "QA-Tf14: QA-Orthotransversal Point". ENCYCLOPEDIA OF QUADRI-FIGURES (EQF). Retrieved 2024-11-02.

[16] "Orthopivotal Cubics". Catalogue of Triangle Cubics.

[17] Gibert, Bernard. "Neuberg Cubics" (PDF).

[18] "K053". Cubic in Triangle Plane.

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Example

Properties

Orthopivotal cubic

Example

See also

Notes

References

External links