How to Find the Orthocenter of a Triangle
The orthocenter is the point where the three altitudes of a triangle intersect. An altitude is a line segment drawn from a vertex perpendicular to the line containing the opposite side. Every triangle has exactly one orthocenter — this calculator finds it directly from the (x, y) coordinates of the three vertices, without needing to draw anything.
Formula and method
Given vertices A(x₁, y₁), B(x₂, y₂), and C(x₃, y₃), the altitude from A is the line through A that is perpendicular to side BC. Two vectors are perpendicular when their dot product is zero, so the altitude from A satisfies (x₃−x₂)(x−x₁) + (y₃−y₂)(y−y₁) = 0. Likewise, the altitude from B satisfies (x₃−x₁)(x−x₂) + (y₃−y₁)(y−y₂) = 0. These are two linear equations in x and y, so rewriting them as A₁x + B₁y = C₁ and A₂x + B₂y = C₂ and solving with Cramer's rule (x = (C₁B₂−C₂B₁)/D, y = (A₁C₂−A₂C₁)/D, where D = A₁B₂−A₂B₁) gives the orthocenter directly — the third altitude is guaranteed to pass through the same point, so only two are needed.
Common sources of error
- Collinear points: if the three points lie on a single straight line (D = 0 above), they do not form a triangle and there is no orthocenter.
- Mixing up coordinates: double-check which pair of numbers belongs to which vertex — swapping an x and y silently produces a valid-looking but wrong answer.
- Assuming the orthocenter is always inside: it is only inside the triangle when the triangle is acute; for obtuse triangles it falls outside the triangle entirely.
Where the orthocenter sits
The orthocenter's location relative to the triangle depends entirely on the triangle's angles: inside for an acute triangle, exactly at the right-angle vertex for a right triangle, and outside the triangle (beyond the side opposite the obtuse angle) for an obtuse triangle. It also sits on the Euler line together with the centroid G and circumcenter O, satisfying H = 3G − 2O.
Applications
The orthocenter shows up in triangle geometry proofs, computer graphics and CAD routines that need perpendicular construction points, surveying and structural layout problems involving perpendicular bracing, and in the study of the Euler line and nine-point circle, which passes through the midpoints of segments from the orthocenter to each vertex.