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

(4)

Other formulas satisfied by the altitude include

(5)

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 ,

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References Bogomolny, A. "The Altitudes." http://www.cut-the-knot.org/triangle/altitudes.html . 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: 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. Referenced on Wolfram|Alpha Altitude
Cite this as:
Weisstein, Eric W. "Altitude." From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/Altitude.html

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