orthocenter

In Euclidean geometry, the orthocenter of a triangle is the common intersection point of its three altitudes, the lines through each vertex perpendicular to the opposite side. It can be found by constructing the perpendiculars from the vertices to the opposite sides and locating their intersection.

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

The orthocenter is related to several other classical centers. The circumcenter, centroid, and orthocenter are collinear on the Euler line, with OG:GH = 1:2, where O is the circumcenter and G is the centroid; equivalently OH = 3·OG. The center of the nine-point circle, N, is the midpoint of OH, and the nine-point circle passes through the feet of the altitudes and the midpoints of the triangle’s sides.

Other notable properties include that the reflections of the orthocenter across the triangle’s sides lie on the circumcircle, and that the feet of the altitudes and the midpoints of the sides lie on the nine-point circle with radius equal to half the circumradius.

Computationally, given triangle vertices A, B, and C, the orthocenter is found as the intersection of the line through A perpendicular to BC and the line through B perpendicular to AC (or equivalently by solving the corresponding perpendicular lines).