Orthocenter Calculator
Enter the x and y coordinates of all three triangle vertices. The calculator finds the orthocenter H (the point where all three altitudes meet), shows its exact coordinates, identifies whether the triangle is acute, right, or obtuse, and walks through every altitude equation so you can follow the working.
Formula
Worked example
A = (0, 0), B = (6, 0), C = (2, 4). Slope of BC = (4-0)/(2-6) = -1. Altitude from A: slope = 1, equation y = x. Slope of AC = (4-0)/(2-0) = 2. Altitude from B: slope = -1/2, equation y - 0 = -0.5(x - 6) => y = -0.5x + 3. Solve x = -0.5x + 3 => x = 2, y = 2. Orthocenter H = (2, 2).
What is the orthocenter of a triangle?
The orthocenter is the single point where all three altitudes of a triangle intersect. An altitude is a line segment drawn from a vertex perpendicular to the opposite side (or the line containing it). Because three altitudes always meet at one point, the orthocenter is called a "point of concurrency." It is one of the four classical triangle centers alongside the centroid, circumcenter, and incenter, and all four lie on the Euler line for any non-equilateral triangle.
How to find the orthocenter step by step
Label the three vertices A(x1, y1), B(x2, y2), and C(x3, y3). First, compute the slope of side BC: m_BC = (y3 - y2) / (x3 - x2). The altitude from A is perpendicular to BC, so its slope is -1 / m_BC, and its equation is y - y1 = (-1/m_BC)(x - x1). Next, compute the slope of side AC and write the altitude from B in the same way. Solve the two altitude equations as a 2x2 linear system. The solution (x_H, y_H) is the orthocenter. Special cases arise when a side is horizontal (altitude is vertical) or vertical (altitude is horizontal), and these are handled by substituting directly rather than using an undefined slope.
Where does the orthocenter lie?
The position of H relative to the triangle depends entirely on the triangle type. For an acute triangle, all interior angles are less than 90 degrees and the orthocenter falls inside the triangle. For a right triangle, the altitude from the right-angle vertex coincides with the two legs, so the orthocenter lands exactly at the right-angle vertex itself. For an obtuse triangle, the altitude from the obtuse-angle vertex must be extended outside the triangle before meeting the other two altitudes, so H lies outside the triangle on the opposite side from the obtuse vertex. For an equilateral triangle, the orthocenter, centroid, circumcenter, and incenter all coincide at the same central point.
The Euler line and triangle centers
In any non-equilateral triangle, the orthocenter H, the centroid G, and the circumcenter O are collinear. This line is called the Euler line. The centroid always lies between H and O, exactly one third of the way from O to H (i.e., OG = (1/3) OH). The nine-point circle passes through the midpoints of all three sides, the feet of all three altitudes, and the midpoints of the three segments from each vertex to H. Because of these relationships, knowing the orthocenter is a key step in fully characterizing a triangle.
Orthocenter position by triangle type
| Triangle type | Defining condition | Orthocenter position |
|---|---|---|
| Acute | All angles < 90 deg | Inside the triangle |
| Right | One angle = 90 deg | At the right-angle vertex |
| Obtuse | One angle > 90 deg | Outside the triangle |
| Equilateral | All sides equal | Center (same as centroid and circumcenter) |
Where H lies relative to the triangle, based on the nature of the interior angles.
Frequently asked questions
What is the orthocenter of a triangle?
The orthocenter is the point where all three altitudes of a triangle intersect. An altitude is drawn from each vertex perpendicular to the opposite side. Because the three altitudes always meet at exactly one point, it is called a point of concurrency and is labeled H by convention.
Can the orthocenter be outside the triangle?
Yes. For an obtuse triangle, at least one altitude must be extended beyond the triangle before it meets the other two, so H lies outside the boundary. For a right triangle, H falls exactly at the vertex holding the right angle. Only for acute triangles does H lie strictly inside.
What happens when two vertices are the same or all three are collinear?
If any two vertices share the same coordinates, or if all three vertices lie on a single straight line, the points do not form a triangle. In that case the altitudes are parallel or undefined and the orthocenter does not exist. This calculator will flag the triangle as degenerate.
How do I handle a vertical or horizontal side?
When a side is vertical (x-coordinates of its endpoints are equal), its slope is undefined, and the perpendicular altitude is horizontal. When a side is horizontal (y-coordinates equal), the altitude is vertical. In both cases the altitude equation simplifies to a single constant coordinate rather than a slope-intercept form. This calculator handles all such cases automatically.
What is the Euler line?
For any non-equilateral triangle, the orthocenter H, centroid G, and circumcenter O are collinear. That collinear relationship is called the Euler line. The centroid always divides the segment from the circumcenter to the orthocenter in a 1:2 ratio, meaning OG = (1/3) OH.
Can I use angle measures instead of coordinates?
Yes, there is a trigonometric formula: x_H = (x1 tan A + x2 tan B + x3 tan C) / (tan A + tan B + tan C), and similarly for y_H, where A, B, C are the interior angles at the respective vertices. However, this requires knowing all three angles, whereas the coordinate method only needs the vertex positions.