Orthocenter
The orthocenter of a triangle is the point of intersection of its altitudes. It is conventionally denoted .
Proof of Existence
Note: The orthocenter's existence is a trivial consequence of the trigonometric version Ceva's Theorem; however, the following proof, due to Leonhard Euler, is much more clever, illuminating and insightful.
Consider a triangle
with circumcenter
and centroid
. Let
be the midpoint of
. Let
be the point such that
is between
and
and
. Then the triangles
,
are similar by angle-side-angle similarity. It follows that
is parallel to
and is therefore perpendicular to
; i.e., it is the altitude from
. Similarly,
,
, are the altitudes from
,
. Hence all the altitudes pass through
. Q.E.D.
This proof also gives us the result that the orthocenter, centroid, and circumcenter are collinear, in that order, and in the proportions described above. The line containing these three points is known as the Euler line of the triangle.
Properties
- The orthocenter is collinear with the circumcenter and de Longchamps point.
- If the orthocenter's triangle is acute, then the orthocenter is on the triangle, it the triangle is right, then it is on the vertex opposite the hypotenuse, and if it is obtuse, then the orthocenter is outside the triangle.
- Let
be a triangle and
its orthocenter. Then the reflections of
over
,
, and
are on the circumcircle of
: