Orthologic triangles

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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]

Two orthologic triangles

Some pairs of orthologic triangles

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The following are some triangles associated with the reference triangle ABC and orthologic with it.[3]

Theorem on orthologic triangles

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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

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Sondat's theorem

References

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  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