Orthocenter Calculator

The orthocenter is the point where all three altitudes of a triangle intersect. An altitude is a perpendicular line from a vertex to the opposite side. The orthocenter is one of the four classical triangle centers.

Find the equations of at least two altitudes (perpendicular from each vertex to the opposite side), then solve the system of equations to find their intersection point. The orthocenter is where all three altitudes meet.

For acute triangles, the orthocenter is inside. For right triangles, it is at the right-angle vertex. For obtuse triangles, it is outside the triangle. This is in contrast to the centroid, which is always inside.

The orthocenter (H), centroid (G), and circumcenter (O) are collinear on the Euler line, with HG = 2GO. The centroid divides the segment from orthocenter to circumcenter in ratio 2:1.

The nine-point circle passes through the midpoints of the sides, the feet of the altitudes, and the midpoints from each vertex to the orthocenter. Its center is the midpoint of the Euler line segment HO.

Yes, for obtuse triangles the orthocenter lies outside the triangle. The altitude from the obtuse angle vertex still falls inside, but the other two altitudes extend outside to meet at the orthocenter.

For a right triangle, the orthocenter coincides with the vertex at the right angle. This is because the legs of the right triangle are already altitudes to each other, so they intersect at the right-angle vertex.

The centroid is the intersection of medians (lines from vertices to midpoints of opposite sides) and always lies inside the triangle. The orthocenter is the intersection of altitudes and can be outside for obtuse triangles.

In vector form: H = A + B + C — 2G, where A, B, C are vertex position vectors and G = (A+B+C)/3 is the centroid. This simplifies to H = A + B + C — 2(A+B+C)/3.

The altitude from vertex A is the line through A perpendicular to side BC. Find the slope of BC (m_BC), then the altitude slope is —1/m_BC. Use point-slope form with vertex A to write the equation.

In an equilateral triangle, the orthocenter, centroid, circumcenter, and incenter all coincide at the same point — the geometric center of the triangle.

The orthocenter is used to prove collinearity (Euler line), compute triangle centers, solve geometry problems, and analyze triangle properties using algebraic methods.

The reflection of the orthocenter over any side of the triangle lies on the circumcircle. Similarly, the reflection over the midpoint of any side also lies on the circumcircle.

Yes, since all three altitudes are concurrent (meet at one point), finding the intersection of any two altitudes gives the orthocenter. The third altitude will automatically pass through the same point.

The distance from the orthocenter to vertex A is |AH| = 2R|cos A|, where R is the circumradius and A is the angle at vertex A. For right triangles, this equals 0 at the right angle.

The reflection of the orthocenter over each side of the triangle lies on the circumcircle. The circumcircle of the triangle formed by the feet of the altitudes is the nine-point circle.

In barycentric coordinates, the orthocenter is H = (tan A : tan B : tan C). This representation is useful for advanced triangle geometry computations.

Given vertices (x₁,y₁), (x₂,y₂), (x₃,y₃), compute the altitude from each vertex as a line equation, then solve two linear equations for their intersection. Use the perpendicular slope and point-slope form.

The Darboux cubic is the locus of points whose pedal triangle has the orthocenter as its symmedian point. It passes through the orthocenter, circumcenter, and several other triangle centers.

The orthocenter is fundamental in triangle geometry, appearing in the Euler line, nine-point circle, and numerous triangle theorems. It connects to trigonometry, coordinate geometry, and advanced Euclidean geometry.

The Orthocenter Calculator uses standard mathematical formulas validated against authoritative references. The specific formula is displayed in the calculator interface with a detailed explanation of each variable.

Each calculator includes a step-by-step worked example showing exactly how the formula is applied. You can follow these steps with pen and paper to verify any result.

The Orthocenter Calculator accepts numeric inputs including integers and decimals. Invalid inputs (letters, special characters) are rejected with clear error messages.