Orthocenter
The orthocenter of a triangle is the point of intersection of its altitudes. It is conventionally denoted as .
Proof of Existence
Note: The orthocenter's existence is a trivial consequence of the trigonometric version of Ceva's Theorem; however, the following proof, due to 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, after the person to whom this proof is due.