Orthologic triangles: Difference between revisions
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[[File:OrthologicTriangles.png|thumb|Two orthologic triangles]] |
[[File:OrthologicTriangles.png|thumb|Two orthologic triangles]] |
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In [[geometry]], two triangles are said to be '''orthologic triangles''' if the perpendiculars from the vertices of one of them to the corresponding sides of the other are concurrent. 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 triangle DEF to the sides BC, CA, AB of triangle ABC are also concurrent. The points of concurrence are known as the '''orthology centres''' of the two triangles.<ref>{{cite web |last1=Weisstein, Eric W. |title=Orthologic Triangles |url=https://mathworld.wolfram.com/OrthologicTriangles.html |website=MathWorld |publisher=MathWorld--A Wolfram Web Resource. |access-date=17 December 2021}}</ref><ref>{{cite book |last1=Gallatly, W. |title=Modern Geometry of the Triangle |date=1913 |publisher=Hodgson, London |pages=55–56 |edition=2 |url=https://archive.org/details/cu31924001522782 |access-date=17 December 2021}}</ref> |
In [[geometry]], two triangles are said to be '''orthologic triangles''' if the perpendiculars from the vertices of one of them to the corresponding sides of the other are concurrent. 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 triangle DEF to the sides BC, CA, AB of triangle ABC are also concurrent. The points of concurrence are known as the '''orthology centres''' of the two triangles.<ref>{{cite web |last1=Weisstein, Eric W. |title=Orthologic Triangles |url=https://mathworld.wolfram.com/OrthologicTriangles.html |website=MathWorld |publisher=MathWorld--A Wolfram Web Resource. |access-date=17 December 2021}}</ref><ref>{{cite book |last1=Gallatly, W. |title=Modern Geometry of the Triangle |date=1913 |publisher=Hodgson, London |pages=55–56 |edition=2 |url=https://archive.org/details/cu31924001522782 |access-date=17 December 2021}}</ref> |
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Revision as of 21:13, 17 December 2021
In geometry, two triangles are said to be orthologic triangles if the perpendiculars from the vertices of one of them to the corresponding sides of the other are concurrent. 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 triangle DEF to the sides BC, CA, AB of triangle 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
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
- ^ 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.