Orthocenter Calculator
Enter three Cartesian vertices to locate the intersection of the triangle's altitudes, classify the triangle, and inspect the geometry on a coordinate plot.
Triangle vertices
Vertex A
Vertex B
Vertex C
Orthocenter results
Orthocenter H
(3.1818, 3.9091)
The triangle is acute, so its orthocenter lies inside the triangle.
Triangle type
Acute
Orthocenter location
Inside triangle
Triangle area
11
Longest side
CA · 6.0828
Enter six valid coordinates for three distinct, non-collinear vertices. Results will update automatically.
Triangle and altitude plot
The plot uses the current coordinate values. Each green altitude passes through a vertex and is perpendicular to the opposite side; their shared intersection is H.
A(1, 1), B(3, 5), C(7, 2), and H(3.1818, 3.9091). H is inside this acute triangle.
The coordinate plot will appear after the triangle is complete and non-collinear.
Coordinate and altitude details
| Item | x / coefficient | y / coefficient | Equation or role |
|---|---|---|---|
| Vertex A | 1 | 1 | Triangle vertex |
| Vertex B | 3 | 5 | Triangle vertex |
| Vertex C | 7 | 2 | Triangle vertex |
| Orthocenter H | 3.1818 | 3.9091 | Intersection of all altitudes |
| Altitude through A | 4 | – 3 | 4x – 3y = 1 |
| Altitude through B | 6 | 1 | 6x + y = 23 |
| Altitude through C | 2 | 4 | 2x + 4y = 22 |
For each altitude, the two displayed coefficients form a vector parallel to the opposite side. The orthocenter satisfies all three equations within floating-point tolerance.
How to use the orthocenter calculator
What this calculator does
This calculator finds the orthocenter H of a non-degenerate triangle from the Cartesian coordinates of vertices A, B, and C. The orthocenter is the single point where the three altitudes meet. An altitude is a line through one vertex that is perpendicular to the opposite side. The tool also classifies the triangle as acute, right, or obtuse; states whether H is inside, on a vertex of, or outside the triangle; calculates the triangle's area and longest side; and plots the vertices, altitude segments, and H on one common coordinate scale. It does not infer missing vertices, measure drawing-scale pixels, or decide whether approximate survey points are geometrically exact.
When to use it
Use it to check coordinate-geometry homework, verify a construction made in dynamic-geometry software, locate the altitude intersection for CAD or analytic-geometry work, or compare how H moves when a triangle changes from acute to right or obtuse. The underlying concurrency theorem is explained in the LibreTexts treatment of altitudes and the orthocenter.
How to calculate
- The calculator opens with a complete demonstration: A = (1, 1), B = (3, 5), and C = (7, 2). Its results and a verified example XLSX workbook are ready immediately.
- Replace the six coordinate values with your own. Results update live; there is no separate Calculate button. Use ordinary decimal notation, with an optional leading minus sign. A dot is the decimal separator. Commas are accepted only in correctly grouped values such as 1,250.5.
- Read Orthocenter H first, then use Triangle type and Orthocenter location to understand whether the intersection should appear inside or outside the triangle. Confirm the geometry in the plot and inspect the altitude equations in the details table.
- Select Download Excel to export the current typed inputs and canonical results. Reset clears the demonstration and all calculated content rather than restoring the sample. After Reset, Download Excel remains disabled until six valid coordinates form a non-collinear triangle again.
Input guide
A x-coordinate (x₁) and A y-coordinate (y₁) are required real numbers defining vertex A; the startup examples are 1 and 1. B x-coordinate (x₂) and B y-coordinate (y₂) define vertex B; the examples are 3 and 5. C x-coordinate (x₃) and C y-coordinate (y₃) define vertex C; the examples are 7 and 2. Each field accepts finite values from – 10¹² through 10¹². Coordinates may be negative or fractional, and all six use the same arbitrary Cartesian unit. Moving any coordinate changes the side directions, so it can move H in either axis and can change the triangle classification. Do not enter scientific notation such as 1e3, a decimal comma such as 1,5, unit text, or three collinear points. Duplicate vertices and nearly collinear inputs are rejected because a unique triangle and orthocenter do not exist.
Output guide
Orthocenter H reports the exact computed intersection as orthocenter-x and orthocenter-y, displayed to four decimal places. Triangle type compares squared side lengths: acute means every angle is below 90°, right means one angle is 90° within numerical tolerance, and obtuse means one angle exceeds 90°. Orthocenter location follows that classification: inside for acute, at the right-angle vertex for right, and outside for obtuse. Triangle area is the absolute coordinate cross-product divided by two; zero would mean collinearity and is therefore invalid. Longest side names AB, BC, or CA and gives its length. The coordinate table lists A, B, C, and H, while the final three rows show altitude equations in the form px + qy = r. The plot encodes the same points and altitude relationships; it is an analytical view, not a separate approximation.
Worked example
With A = (1, 1), B = (3, 5), and C = (7, 2), the altitude through A is perpendicular to BC. Because the vector from B to C is (4, – 3), that altitude satisfies 4(x – 1) – 3(y – 1) = 0, or 4x – 3y = 1. The altitude through B is perpendicular to AC, whose vector is (6, 1), so 6(x – 3) + (y – 5) = 0, or 6x + y = 23. Solving those two linear equations gives x = 35/11 = 3.1818 and y = 43/11 = 3.9091. The third altitude equation, 2x + 4y = 22, is satisfied by the same point. The displayed area is 11 square units, the longest side is CA at 6.0828 units, and the triangle is acute, placing H inside. For a concise reference definition and additional properties, see Wolfram MathWorld's orthocenter entry.
How the coordinate formula works
The calculation avoids slope formulas, so vertical and horizontal sides need no special-case division. Let H = (x, y). A line from A to H is perpendicular to side BC exactly when their direction vectors have dot product zero. This produces one linear equation. A second equation comes from the altitude through B:
(H – A) · (C – B) = 0
(H – B) · (C – A) = 0
The two equations are solved as a 2 × 2 linear system. A nonzero determinant is equivalent to a nonzero triangle area, so the same test both prevents division by zero and rejects collinear vertices. The third altitude is then evaluated as an independent consistency check. Right-triangle behavior also follows directly: the two legs are already altitudes, so H equals the right-angle vertex. The connection between perpendicular components and the Pythagorean theorem is reviewed in OpenStax's right-triangle and distance discussion.
Interpretation and common mistakes
An orthocenter outside the triangle is not an error; it is the expected result for an obtuse triangle because at least two altitude lines meet beyond the side segments. Very large coordinate magnitudes paired with extremely tiny areas can be numerically unstable, so the calculator rejects nearly collinear configurations relative to their scale. Remember that coordinates must share one axis system and one unit. Swapping only one x or y value changes the triangle rather than merely relabeling a point. Finally, rounded display values may not satisfy the altitude equations perfectly if copied back at four decimals; the workbook stores the higher-precision canonical numbers used by the model.