orthologic triangles
{{Short description|Type of symmetry between two triangles}}
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 {{mvar|A, B, C}} of triangle {{math|△ABC}} to the sides {{mvar|EF, FD, DE}} of triangle {{math|△DEF}} are concurrent then the perpendiculars from the vertices {{mvar|D, E, F}} of {{math|△DEF}} to the sides {{mvar|BC, CA, AB}} of {{math|△ABC}} are also concurrent. The points of concurrence are known as the orthology centres of the two triangles.{{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}}{{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}}
Some pairs of orthologic triangles
The following are some triangles associated with the reference triangle ABC and orthologic with it.{{cite web |last1=Smarandache, Florentin and Ion Patrascu |title=THE GEOMETRY OF THE ORTHOLOGICAL TRIANGLES |url=https://digitalrepository.unm.edu/math_fsp/260 |access-date=17 December 2021}}
- Medial triangle
- Anticomplementary 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 (and as a special case the Orthic_triangle)