Orthologic triangles explained
In geometry, two triangles are said to be orthologic if the perpendiculars from the vertices of one of them to the corresponding sides of the other are concurrent (i.e., they intersect at a single point). This is a symmetric property; that is, if the perpendiculars from the vertices of triangle to the sides of triangle are concurrent then the perpendiculars from the vertices of to the sides of are also concurrent. The points of concurrence are known as the orthology centres of the two triangles.[1] [2]
Some pairs of orthologic triangles
The following are some triangles associated with the reference triangle ABC and orthologic with it.[3]
- Medial triangle
- Anticomplementary triangle
- Orthic triangle
- The triangle whose vertices are the points of contact of the incircle with the sides of ABC
- Tangential triangle
- The triangle whose vertices are the points of contacts of the excircles with the respective sides of triangle ABC
- The triangle formed by the bisectors of the external angles of triangle ABC
- The pedal triangle of any point P in the plane of triangle ABC
Theorem on orthologic triangles
Sondat's theorem states that If two triangles ABC and A'B'C' are perspective and orthologic, then the center of perspective P and the orthologic centers Q and Q' are on the same line perpendicular to the axis of perspectivity [4]
See also
Sondat's theorem
Notes and References
- Web site: Weisstein, Eric W. . Orthologic Triangles . MathWorld . MathWorld--A Wolfram Web Resource. . 17 December 2021.
- Book: Gallatly, W. . Modern Geometry of the Triangle . 1913 . Hodgson, London . 55–56 . 2 . 17 December 2021.
- Web site: Smarandache, Florentin and Ion Patrascu . THE GEOMETRY OF THE ORTHOLOGICAL TRIANGLES . 17 December 2021.
- https://ijgeometry.com/wp-content/uploads/2011/10/21.pdf Ion Patrascu and Catalin Barbu, Two new proof of Goormaghtigh's theorem, International journal of geometry, Vol. 1 (2012), No. 1, 10 - 19
