| 释义 |
AltitudeThe altitudes of a Triangle are the Cevians  which are Perpendicular to the Legs  opposite  . They have lengths  given by
 | (1) |
 | (2) |
 is the Semiperimeter of  and  . Another interesting Formula is
 | (3) |
 is the Area of the Triangle and  is theSemiperimeter of the altitude triangle  . The three altitudes of any Triangle areConcurrent at the Orthocenter  . This fundamental fact did not appear anywhere inEuclid's  Elements">Elements.
Other formulas satisfied by the altitude include
 | (4) |
 | (5) |
 | (6) |
 is the Inradius and  are the Exradii (Johnson 1929, p. 189). In addition,
 | (7) |
 | (8) |
 is the Circumradius.
The points  ,  ,  , and  (and their permutations with respect to indices) all lie on a Circle, asdo the points , , , and (and their permutations with respect to indices). Triangles and  are inversely similar.
The triangle  has the minimum Perimeter of any Triangle inscribed in a given Acute Triangle(Johnson 1929, pp. 161-165). The Perimeter of is  (Johnson 1929, p. 191). Additionalproperties 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. |
