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Altitude


Altitudes

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

h_a=(bc)/(2R)=csinB=bsinC
(1)
h_b=(ac)/(2R)=asinC=csinA
(2)
h_c=(ab)/(2R)=bsinA=asinB,
(3)

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

 h_ah_bh_c=((abc)^2)/(8R^3).
(4)

Other formulas satisfied by the altitude include

 1/(h_1)+1/(h_2)+1/(h_3)=1/r,
(5)

where r is the inradius, and

1/(r_1)=1/(h_2)+1/(h_3)-1/(h_1)
(6)
1/(r_2)+1/(r_3)=1/r-1/(r_1)
(7)
=2/(h_1),
(8)

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

HA_1·HH_1=HA_2·HH_2
(9)
=HA_3·HH_3
(10)
=1/2(a_1^2+a_2^2+a_3^2)-4R^2,
(11)

where R is again the circumradius.

AltitudeCircles

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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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.

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Altitude

Cite this as:

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

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