The altitudes of a triangle are the Cevians A_iH_i that are perpendicular to the legs A_jA_k opposite A_i. The three altitudes of any triangle are concurrent at the orthocenter H (Durell 1928). This fundamental fact did not appear anywhere in Euclid's Elements.

The triangle DeltaH_1H_2H_3 connecting the feet of the altitudes is known as the orthic triangle.

The altitudes of a triangle with side length a, b, and c and vertex angles A, B, C have lengths given by


where R is the circumradius of DeltaABC. This leads to the beautiful formula


Other formulas satisfied by the altitude include


where r is the inradius, and


where r_i are the exradii (Johnson 1929, p. 189). In addition,


where R is again the circumradius.


The points A_1, A_3, H_1, and H_3 (and their permutations with respect to indices; left figure) all lie on a circle, as do the points A_2, H_3, H, and H_1 (and their permutations with respect to indices; right figure).

Triangles DeltaA_1A_2A_3 and DeltaA_1H_2H_3 are inversely similar.

Additional properties involving the feet of the altitudes are given by Johnson (1929, pp. 261-262). The line joining the feet to two altitudes of a triangle is antiparallel to the third side (Johnson 1929, p. 172).

See also

Cevian, Maltitude, Orthic Triangle, Orthocenter, Perpendicular, Perpendicular Foot, Taylor Circle Explore this topic in the MathWorld classroom

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Bogomolny, A. "The Altitudes.", H. S. M. and Greitzer, S. L. "More on the Altitude and Orthocentric Triangle." §2.4 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 9 and 36-40, 1967.Durell, C. V. Modern Geometry: The Straight Line and Circle. London: Macmillan, p. 20, 1928.Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, 1929.

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Cite this as:

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

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