Orthocenter Calculator

How it works

H is the intersection of the altitudes: (H − A)·(C − B) = 0 and (H − B)·(C − A) = 0, solved as a 2 × 2 linear system · altitude from A: line through A perpendicular to BC (slope −1/m_BC) · altitude length h_A = 2·Area / |BC| · Euler line: H = 3G − 2O

An altitude is the segment from a vertex perpendicular to the opposite side, and the three altitudes of any triangle meet at one point, the orthocenter. The calculator writes two of the perpendicularity conditions as dot products, solves them exactly with fractions, and lists all three altitude lines so you can check each one passes through H. The orthocenter lies inside an acute triangle, at the right-angle vertex of a right triangle and outside an obtuse triangle. It also sits on the Euler line with the centroid G and circumcenter O, with H = 3G − 2O (Wolfram MathWorld, Orthocenter).

Worked example

A(0, 0), B(6, 0), C(2, 4). AB is horizontal, so the altitude from C is the vertical line x = 2. BC has slope (4 − 0)/(2 − 6) = −1, so the altitude from A has slope 1: y = x. They meet at H = (2, 2), and the altitude from B, y = −(1/2)x + 3, also passes through (2, 2). All angles are acute, so H is inside the triangle. Centroid G = (8/3, 4/3) and circumcenter O = (3, 1).

Sources: Wolfram MathWorld — Orthocenter · Khan Academy — Altitudes and the orthocenter

Frequently asked questions