Altitude
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 classroomExplore with Wolfram|Alpha
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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
AltitudeCite this as:
Weisstein, Eric W. "Altitude." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Altitude.html