Orthotransversal

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

Orthotransversal

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

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

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).^[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