Orthocenter Calculator

The orthocenter is the common intersection of a triangle's three altitudes. An altitude is a line through one vertex that is perpendicular to the line containing the opposite side. Although drawing all three altitudes makes the construction easy to recognize, only two are needed for calculation: once two distinct altitude lines meet, the third passes through the same point. This concurrency is a fundamental theorem of Euclidean geometry. The orthocenter's location depends on the triangle's angles. In an acute triangle, all three altitudes meet inside the triangle. In a right triangle, the two legs are already perpendicular altitudes, so the orthocenter is the right-angle vertex. In an obtuse triangle, the altitudes intersect outside the triangle because at least two opposite sides must be extended. An exterior coordinate is therefore a valid answer rather than evidence that the calculation failed. Coordinate geometry turns perpendicularity into dot products. A line from vertex A to the unknown point H is an altitude when vector H minus A is perpendicular to side BC. Their dot product must equal zero. A second equation says vector H minus B is perpendicular to side AC. Expanding those two dot products produces a two-by-two linear system in the x- and y-coordinates of H. The calculator solves that system directly, avoiding slope formulas that require special cases for vertical lines. If the three supplied points are collinear, they do not define a triangle. The two altitude equations then fail to have one unique intersection, which appears algebraically as a zero determinant. The calculator detects this condition and requests noncollinear vertices. Repeated points are rejected for the same geometric reason. Decimal and negative coordinates are accepted, so the method works in every quadrant and does not require axis-aligned sides. The orthocenter is one of four classical triangle centers. The centroid is the intersection of medians, the incenter is the intersection of angle bisectors, and the circumcenter is the intersection of perpendicular bisectors. In a general triangle these are different points. The centroid, circumcenter, and orthocenter lie on the Euler line, and their positions obey a simple ratio. In an equilateral triangle, symmetry causes all four principal centers to coincide. This calculator is useful for analytic geometry, checking graph constructions, triangle-center exercises, and computer graphics. Enter each ordered pair carefully, preserving which x-coordinate belongs with which y-coordinate. The result is rounded for readable display while the equations are evaluated with full JavaScript floating-point precision. Substituting the result into either altitude dot-product equation provides a direct way to verify it.