orthocenters

The orthocenter of a triangle is the common intersection point of its three altitudes, the perpendiculars from each vertex to the opposite side (or its extension). In triangle ABC, the altitudes from A, B, and C meet at the orthocenter H.

The position of the orthocenter depends on the triangle’s type. In an acute triangle, H lies inside the triangle. In a right triangle, H coincides with the vertex at the right angle. In an obtuse triangle, H lies outside the triangle.

Several important relations connect the orthocenter to other centers. The circumcenter O (center of the circumscribed circle) and the centroid G (intersection of the medians) lie on the Euler line with H; specifically, O, G, and H are collinear and OG:GH = 1:2, with G between O and H. The nine-point center N is the midpoint of OH, and the nine-point circle passes through the midpoints of the sides and the feet of the altitudes. An oft-cited distance relation uses the circumradius R: AH = 2R cos A, BH = 2R cos B, CH = 2R cos C. This implies H is inside for acute triangles (cosines positive) and outside for obtuse ones (one cosine negative).

A classical geometric property is that the reflection of H across any side of the triangle lies on the circumcircle. In vector terms, if the circumcenter is taken as the origin and the vertices as position vectors a, b, c on the circumcircle, then the orthocenter’s vector is h = a + b + c.