Orthocenter

The orthocenter of a triangle is the point of intersection of its altitudes. It is conventionally denoted as $\displaystyle H$ .

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 $\displaystyle ABC$ with circumcenter $\displaystyle O$ and centroid $\displaystyle G$ . Let $\displaystyle A'$ be the midpoint of $\displaystyle BC$ . Let $\displaystyle H$ be the point such that $\displaystyle G$ is between $\displaystyle H$ and $\displaystyle O$ and $\displaystyle HG = 2 HO$ . Then the triangles $\displaystyle AGH$ , $\displaystyle A'GO$ are similar by angle-side-angle similarity. It follows that $\displaystyle AH$ is parallel to $\displaystyle OA'$ and is therefore perpendicular to $\displaystyle BC$ ; i.e., it is the altitude from $\displaystyle A$ . Similarly, $\displaystyle BH$ , $\displaystyle CH$ , are the altitudes from $\displaystyle B$ , $\displaystyle {C}$ . Hence all the altitudes pass through $\displaystyle H$ . 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.

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Orthocenter

Proof of Existence