Orthocenter - Triangle Altitude Intersection

An orthocenter calculator returns the point where the three altitudes of a triangle meet, given three vertex coordinates. The altitudes are the lines through each vertex that are perpendicular to the opposite side, and they all cross at one point, the orthocenter, which is one of the four classical triangle centers. The orthocenter equals the intersection of the altitude from A perpendicular to BC and the altitude from B perpendicular to AC, and the same point also lies on the third altitude from C.

• • Coordinate geometry homework: Verify textbook exercises where the triangle is given as three coordinate points and the orthocenter must be reported in the same length unit as the input grid.
• • Classifying a triangle by its orthocenter location: The orthocenter sits inside an acute triangle, on the right-angle vertex of a right triangle, and outside an obtuse triangle, so the orthocenter position is a quick way to label the triangle type.
• • Pre-step for Euler line and nine-point circle work: The orthocenter, centroid, and circumcenter are collinear on the Euler line in a fixed 2:1 ratio, so the orthocenter is the first triangle center needed before computing the nine-point center.
• • CAD and graphics triangle centers: Mark the orthocenter of a triangle stored in a model as three vertex coordinates, especially when the triangle is rotated away from the axes and a base-height shortcut does not apply.

A triangle in the plane is uniquely defined by three non-collinear points, so its orthocenter is a function of six coordinates. The same six coordinates also feed the centroid, but the two centers coincide only for an equilateral triangle.

For the same three vertex coordinates the Centroid Calculator returns the arithmetic mean of the three vertices, which is the second of the four classical triangle centers.

The orthocenter calculator writes the altitude from A as the line through A perpendicular to BC, writes the altitude from B as the line through B perpendicular to AC, and solves the two-line intersection with Cramer's rule. The third-altitude cross-check confirms the result.

• x1, y1, x2, y2, x3, y3: Cartesian coordinates of the three triangle vertices, in the same length unit.
• Hx, Hy: Orthocenter coordinates returned by the two-altitude intersection, in the same length unit as the input coordinates.

The same six coordinates that produce the orthocenter also produce the triangle area through the shoelace formula, so the area-triangle-coordinates-calculator is the natural next step when both the center and the area of a triangle are needed in the same report.

According to Wolfram MathWorld, the orthocenter of a triangle is the point of concurrency of the three altitudes, and in closed form the orthocenter coordinates are H = (x1 + x2 + x3 - 2*Ox, y1 + y2 + y3 - 2*Oy), where (Ox, Oy) is the circumcenter of the same triangle.

The same six coordinates that produce the orthocenter also produce the triangle area through the shoelace formula, so the Area Triangle Coordinates Calculator is the natural next step when both the center and the area of a triangle are needed in the same report.

Four ideas decide what the orthocenter means for a triangle and how the formula behaves on edge cases.

For any non-equilateral triangle the orthocenter, centroid, and circumcenter are collinear on the Euler line, with the centroid sitting two-thirds of the way from the circumcenter to the orthocenter (HG:GO = 2:1), while the incenter does not lie on this line in general. For a right triangle the orthocenter is the right-angle vertex and the circumcenter is the midpoint of the hypotenuse, so the two centers sit at different points. The output is in the same length unit as the input coordinates, so the page does not convert units.

When all three angles of the triangle are acute the orthocenter sits inside the figure, and the Acute Triangle explores the same acute case with the angle sum and side length bounds that the orthocenter location depends on.

Use the orthocenter calculator with all six coordinates in the same length unit, then read the orthocenter (Hx, Hy), the third-altitude cross-check, and the triangle type in the same pass.

1. 1 Enter the first vertex: Type the x and y coordinates of the first vertex. Use the same length unit for both coordinates and for the other two vertices so the orthocenter comes out in that unit.
2. 2 Enter the second and third vertices: Type the x and y coordinates of the second and third vertices. The order of the vertices does not change the orthocenter because the formula is symmetric in the three inputs.
3. 3 Read the orthocenter: Use the orthocenter point in the black results panel as the primary answer. Hx and Hy are also listed below as separate numeric rows in the same length unit as the inputs.
4. 4 Cross-check with the third altitude: The page shows the third-altitude cross-check, which is the orthocenter recomputed from the altitude from C perpendicular to AB. The two values match for any non-degenerate triangle.
5. 5 Check the right-triangle case: When the triangle is a right triangle, the orthocenter coincides with the right-angle vertex. The right-triangle calculator on this site is the natural pair when the input comes from a right triangle, because it shares the same length unit and the orthocenter lands at the right-angle vertex.

For a triangle with vertices (0, 0), (6, 0), and (0, 9), the orthocenter is (0, 0). The result panel reports Hx = 0 and Hy = 0, the third-altitude cross-check matches, and the triangle type label reads 'Right triangle'.

For a right triangle the orthocenter lands at the right-angle vertex, which is a useful shortcut the Right Triangle Calculator explores in more depth with the same length unit and the same six-coordinate input pattern.

The two-altitude intersection is a short calculation, but the orthocenter calculator wraps it in a layout that helps with plotting, verifying, and reporting the orthocenter of a triangle.

• • Direct from six coordinates: The two-altitude intersection takes the three vertices straight to the orthocenter, so no medians, side lengths, or perpendicular heights are needed first.
• • Orthocenter point is shown explicitly: The page formats the orthocenter as (Hx, Hy) so it can be plotted on a coordinate plane or pasted into a report without further formatting.
• • Third-altitude cross-check is included: The cross-check recomputes the orthocenter from the third altitude, so a single page can confirm the two-altitude intersection and the third altitude at the same time.
• • Triangle type label is built in: The page reports the triangle as right, obtuse, acute, or degenerate, so the user knows immediately whether the orthocenter is on, inside, or outside the triangle.

The orthocenter calculator returns the orthocenter point, the two coordinate rows, the third-altitude cross-check, and the triangle type label together.

When the input is given as side lengths or a base and a height instead of three vertex coordinates, the Triangle Calculator handles the SSS, SAS, and base-height cases in the same Math and Conversion cluster.

The two-altitude intersection is stable, but a few input choices decide whether the orthocenter calculator returns the right value for a given triangle.

• • The calculator does not solve for a missing vertex when only the orthocenter and two vertices are known. Use a system of equations for that case instead.
• • Hand calculation that rounds the intermediate slopes partway through can differ by a few hundredths of a length unit from the calculator, which keeps full precision until the display step.
• • For polygons with more than three sides the orthocenter is not defined, and the triangle-area-calculator is the natural next step when the shape is a polygon instead of a triangle.

For an obtuse triangle the orthocenter sits outside the triangle, so a plot on a coordinate plane will need extra room around the triangle. For an equilateral triangle the orthocenter coincides with the centroid, incenter, and circumcenter.

According to Wikipedia, the orthocenter of a triangle is the point where the three altitudes meet, and for a right triangle it coincides with the vertex of the right angle.

For polygons with more than three sides the orthocenter is not defined, and the Triangle Area Calculator is the natural next step when the shape is a polygon instead of a triangle.