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

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

The altitudes of a triangle with side length , , and and vertex angles , , have lengths given by

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

Other formulas satisfied by the altitude include

where is the inradius, and

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

where is again the circumradius.

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

Triangles and 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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References

Bogomolny, A. "The Altitudes." https://www.cut-the-knot.org/triangle/altitudes.shtml.Coxeter, 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, England: 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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Altitude

Cite this as:

Weisstein, Eric W. "Altitude." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Altitude.html

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