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# Orthic Triangle

Given a triangle , the triangle whose vertices are endpoints of the altitudes from each of the vertices of is called the orthic triangle, or sometimes the altitude triangle. The three lines , , and are concurrent at the orthocenter of .

The orthic triangle is therefore both the pedal triangle and Cevian triangle with respect to (Kimberling 1998, p. 156). It is also the cyclocevian triangle of the triangle centroid .

 (1)

The area of the orthic triangle is given by

 (2)

where is the circumradius of .

The orthic triangle has the minimum perimeter of any triangle inscribed in a given acute triangle (Johnson 1929, pp. 161-165). The lengths of the legs of the orthic triangle are given by

 (3) (4) (5)

The inradius of the orthic triangle is

 (6)

where is the circumradius of the reference triangle (Johnson 1929, p. 191), and the circumradius is

 (7)

For an obtuse triangle or right triangle, the semiperimeter is

 (8)

which simplifies in the case of an acute triangle to

 (9)

where is the triangle area of and(Johnson 1929, p. 191).

Given a triangle , construct the orthic triangle and determine the symmedian points , , and of , , and , respectively. Then the -symmedian of the corner triangle is the -median of triangle , and similarly for the (Honsberger 1995, p. 75). In addition, the -median of the corner triangle is the -symmedian of triangle , and similarly for the other two corner triangles.

Finally, the Euler lines of the three corner triangles , and pass through the Euler points, and concur at a point on the nine-point circle of triangle such that one of the following holds

 (10) (11) (12)

(Thébault 1947, 1949; Thébault et al. 1951).

The sides of the orthic triangle are parallel to the tangents to the circumcircle at the vertices (Johnson 1929, p. 172). This is equivalent to the statement that each line from a triangle's circumcenter to a vertex is always perpendicular to the corresponding side of the orthic triangle (Honsberger 1995, p. 22), and to the fact that the orthic and tangential triangles are homothetic at Kimberling center .

The triangle centroid of the orthic triangle has triangle center function

 (13)

(Casey 1893, Kimberling 1994), which is Kimberling center . The symmedian point of the orthic triangle has triangle center function

 (14)

(Casey 1893, Kimberling 1994), which is Kimberling center .

The following table gives the centers of the orthic triangle in terms of the centers of the reference triangle that correspond to Kimberling centers .

 center of orthic triangle center of reference triangle incenter radical center of (circumcircle, Parry circle, Bevan circle) triangle centroid triangle centroid of orthic triangle circumcenter nine-point center orthocenter orthocenter of orthic triangle nine-point center nine-point center of orthic triangle symmedian point symmedian point of orthic triangle Euler infinity point isogonal conjugate of -of-orthic-triangle Tarry point -of-orthic-triangle Steiner point| -of-orthic-triangle psi(symmedian point, orthocenter) -of-orthic-triangle focus of Kiepert parabola -of-orthic-triangle Parry point -of-orthic-triangle psi(orthocenter, symmedian point) -of-orthic-triangle Parry reflection point isogonal conjugate of isogonal conjugate of Napoleon crossdifference

Altitude, Fagnano's Problem, Orthic Inconic, Orthocenter, Pedal Triangle, Symmedian Point

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## References

Casey, J. A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing an Account of Its Most Recent Extensions, with Numerous Examples, 2nd ed., rev. enl. Dublin: Hodges, Figgis, & Co., p. 9, 1893.Coxeter, H. S. M. and Greitzer, S. L. "The Orthic Triangle." §1.6 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 9 and 16-18, 1967.Honsberger, R. "The Orthic Triangle." §2.3 in Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 21-25, 1995.Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, 1929.Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163-187, 1994.Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.Thébault, V. "Concerning the Euler Line of a Triangle." Amer. Math. Monthly 54, 447-453, 1947.Thébault, V. "Problem 4328." Amer. Math. Monthly 56, 39-40, 1949.Thébault, V.; Ramler, O. J.; and Goormaghtigh, R. "Solution to Problem 4328: Euler Lines." Amer. Math. Monthly 58, 45, 1951.

Orthic Triangle

## Cite this as:

Weisstein, Eric W. "Orthic Triangle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/OrthicTriangle.html