Orthic Triangle


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

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

Its trilinear vertex matrix is

 [0 secB secC; secA 0 secC; secA secB 0].

The area of the orthic triangle is given by


where R is the circumradius of DeltaABC.

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


The inradius of the orthic triangle is


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


For an obtuse triangle or right triangle, the semiperimeter is

 s_H={acosBcosC   if A>=1/2pi; bcosCcosA   if B>=1/2pi; ccosAcosB   if C>=1/2pi,

which simplifies in the case of an acute triangle to


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


Given a triangle DeltaA_1A_2A_3, construct the orthic triangle DeltaH_1H_2H_3 and determine the symmedian points K_1, K_2, and K_3 of DeltaA_1H_2H_3, DeltaH_1A_2H_3, and DeltaH_1H_2A_3, respectively. Then the A_1-symmedian of the corner triangle DeltaA_1H_2H_3 is the A_1-median of triangle DeltaA_1A_2A_3, and similarly for the (Honsberger 1995, p. 75). In addition, the A_1-median of the corner triangle DeltaA_1H_2H_3 is the A_1-symmedian of triangle DeltaA_1A_2A_3, and similarly for the other two corner triangles.


Finally, the Euler lines of the three corner triangles DeltaA_1H_2H_3, DeltaA_2H_3H_1 and DeltaA_3H_1H_2 pass through the Euler points, and concur at a point P on the nine-point circle of triangle DeltaA_1A_2A_3 such that one of the following holds


(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 X_(25).

The triangle centroid of the orthic triangle has triangle center function


(Casey 1893, Kimberling 1994), which is Kimberling center X_(51). The symmedian point of the orthic triangle has triangle center function


(Casey 1893, Kimberling 1994), which is Kimberling center X_(53).

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

X_ncenter of orthic triangleX_ncenter of reference triangle
X_1incenterX_(2503)radical center of (circumcircle, Parry circle, Bevan circle)
X_2triangle centroidX_(51)triangle centroid of orthic triangle
X_3circumcenterX_5nine-point center
X_4orthocenterX_(52)orthocenter of orthic triangle
X_5nine-point centerX_(143)nine-point center of orthic triangle
X_6symmedian pointX_(53)symmedian point of orthic triangle
X_(30)Euler infinity pointX_(1154)isogonal conjugate of X_(1141)
X_(98)Tarry pointX_(129)X_(98)-of-orthic-triangle
X_(99)Steiner point|X_(130)X_(99)-of-orthic-triangle
X_(107)psi(symmedian point, orthocenter)X_(134)X_(107)-of-orthic-triangle
X_(110)focus of Kiepert parabolaX_(137)X_(110)-of-orthic-triangle
X_(111)Parry pointX_(138)X_(111)-of-orthic-triangle
X_(112)psi(orthocenter, symmedian point)X_(139)X_(112)-of-orthic-triangle
X_(399)Parry reflection pointX_(1263)isogonal conjugate of X_(1157)
X_(523)isogonal conjugate of X_(110)X_(1510)Napoleon crossdifference

See also

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

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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.

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Orthic Triangle

Cite this as:

Weisstein, Eric W. "Orthic Triangle." From MathWorld--A Wolfram Web Resource.

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