Altitude

The altitudes of a Triangle are the Cevians $A_iH_i$ which are Perpendicular to the Legs $A_jA_k$ opposite $A_i$. They have lengths $h_i\equiv \overline{A_iH_i}$ given by

$$h_i=a_{i+1}\sin\alpha_{i+2} = a_{i+2}\sin\alpha_{i+1}$$ (1)

$$h_1 ={2\sqrt{s(s-a_1)(s-a_2)(s-a_3)}\over a_1},$$ (2)

where $s$ is the Semiperimeter of $\Delta A_1A_2A_3$ and $a_i\equiv\overline{A_jA_k}$. Another interesting Formula is

$$h_1h_2h_3=2s_h\Delta$$ (3)

(Johnson 1929, p. 191), where $\Delta$ is the Area of the Triangle $\Delta A_1A_2A_3$ and $s_h$ is the Semiperimeter of the altitude triangle $\Delta H_1H_2H_3$. The three altitudes of any Triangle are Concurrent at the Orthocenter $H$. This fundamental fact did not appear anywhere in Euclid's Elements.

Other formulas satisfied by the altitude include

$${1\over h_1}+{1\over h_2}+{1\over h_3}={1\over r}$$ (4)

$${1\over r_1}={1\over h_2}+{1\over h_3}-{1\over h_1}$$ (5)

$${1\over r_2}+{1\over r_3}={1\over r}-{1\over r_1}={2\over h_1},$$ (6)

where $r$ is the Inradius and $r_i$ are the Exradii (Johnson 1929, p. 189). In addition,

$$HA_1\cdot HH_1 = HA_2\cdot HH_2 = HA_3\cdot HH_3$$ (7)

$$HA_1\cdot HH_1 = {\textstyle{1\over 2}}({a_1}^2+{a_2}^2+{a_3}^2)-4R^2,$$ (8)

where $R$ is the Circumradius.

The points $A_1$, $A_3$, $H_1$, and $H_3$ (and their permutations with respect to indices) all lie on a Circle, as do the points $A_3$, $H_3$, $H$, and $H_1$ (and their permutations with respect to indices). Triangles $\Delta A_1A_2A_3$ and $\Delta A_1H_2H_3$ are inversely similar.

The triangle $H_1H_2H_3$ has the minimum Perimeter of any Triangle inscribed in a given Acute Triangle (Johnson 1929, pp. 161-165). The Perimeter of $\Delta H_1H_2H_3$ is $2\Delta/R$ (Johnson 1929, p. 191). Additional properties involving the Feet of the altitudes are given by Johnson (1929, pp. 261-262).

See also Cevian, Foot, Orthocenter, Perpendicular, Perpendicular Foot

References

Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 9 and 36-40, 1967.

Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, 1929.