Orthologic triangles

Two orthologic 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 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

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]: Thm. 1.6 

See also

Sondat's theorem

References

1. Weisstein, Eric W. "Orthologic Triangles". MathWorld. MathWorld--A Wolfram Web Resource. Retrieved 17 December 2021.
2. Gallatly, W. (1913). Modern Geometry of the Triangle (2 ed.). Hodgson, London. pp. 55–56. Retrieved 17 December 2021.
3. Smarandache, Florentin and Ion Patrascu. "THE GEOMETRY OF THE ORTHOLOGICAL TRIANGLES". Retrieved 17 December 2021.
4. Ion Patrascu and Catalin Barbu, Two new proof of Goormaghtigh's theorem, International journal of geometry, Vol. 1 (2012), No. 1, 10 - 19 ISSN 2247-9880