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 A, B, C of triangle △ABC to the sides EF, FD, DE of triangle △DEF are concurrent then the perpendiculars from the vertices D, E, F of △DEF to the sides BC, CA, AB of △ABC are also concurrent. The points of concurrence are known as the orthology centres of the two triangles.[1][2]
Some pairs of orthologic triangles edit
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
References edit
- ^ Weisstein, Eric W. "Orthologic Triangles". MathWorld. MathWorld--A Wolfram Web Resource. Retrieved 17 December 2021.
- ^ Gallatly, W. (1913). Modern Geometry of the Triangle (2 ed.). Hodgson, London. pp. 55–56. Retrieved 17 December 2021.
- ^ Smarandache, Florentin and Ion Patrascu. "THE GEOMETRY OF THE ORTHOLOGICAL TRIANGLES". Retrieved 17 December 2021.