The Compendium Geometry is an eBook providing facts, formulas and explanations about geometry.

Altitudes of a Triangle

The heights (or altitudes) ha, hb, and hc of a triangle are the perpendiculars through the corner points to the opposite sides. All three altitudes intersect at the orthocenter H. The position of H depends on the form of of the triangle:

kind of triangle position of H
acute triangle within the triangle
right triangle on the corner of the right angle
obtuse triangle outside the triangle

The following equations are valid for the relations between the altitudes, the angles and sides:

ha = bsin γ = csin β
hb = csin α = asin γ
hc = asin β = bsin α

There are a two theorems concerning the altitudes of a triangle which can be used to construct triangles:

  • The altitude of a triangle dissects the triangle into two right triangles. The base point of the altitude is always located on the two circles defined by Thales' Theorem.
  • The lengths of the altitudes of a triangle are indirect proportional to the corresponding sides:
    ha : hb = b : a
    ha : hc = c : a
    hb : hc = c : b

    ha : hb : hc = 1/a : 1/b : 1/c

Last Update: 2011-01-11