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Triangle Orthocenter Calculator

Calculate the orthocenter of any triangle by entering the x and y coordinates of all three vertices. Returns Hx and Hy instantly.

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Orthocenter Coordinate / Distance

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Orthocenter Coordinate / Distance—units

The formula

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What Is the Orthocenter of a Triangle?

The orthocenter is the unique point where all three altitudes of a triangle intersect. An altitude is a line segment drawn from each vertex perpendicular to the opposite side — or to the extension of that side in obtuse triangles. According to Wolfram MathWorld, the orthocenter is one of the four classical triangle centers, alongside the centroid, circumcenter, and incenter. Unlike the centroid, the orthocenter does not always lie inside the triangle, making its location a key diagnostic of triangle type.

Formula Derivation

Finding the orthocenter algebraically requires solving a system of two linear equations representing two of the three altitudes. Because all three altitudes are concurrent, only two are needed. The coordinate-based approach introduces six auxiliary variables to keep the computation organized:

  • a = x2 - x3
  • b = y2 - y3
  • c = x1 - x3
  • d = y1 - y3
  • e = x1 times a + y1 times b
  • f = x2 times c + y2 times d

These substitutions reduce the altitude-intersection problem to a 2x2 linear system solvable by Cramer's rule. The resulting orthocenter coordinates H(Hx, Hy) are:

Hx = (e * d - b * f) / (a * d - b * c)

Hy = (a * f - e * c) / (a * d - b * c)

The denominator a * d - b * c is the determinant of the coefficient matrix formed by the altitude direction vectors. A determinant of zero means the three vertices are collinear — no valid triangle exists — and the calculator will flag this as an undefined result. This derivation follows the systems-of-linear-equations approach documented by Cerritos College Department of Mathematics.

Variable Reference

  • x1, y1 — Cartesian coordinates of vertex A
  • x2, y2 — Cartesian coordinates of vertex B
  • x3, y3 — Cartesian coordinates of vertex C
  • Hx — x-coordinate of the orthocenter H
  • Hy — y-coordinate of the orthocenter H

Worked Example

Consider a triangle with vertices A(0, 0), B(4, 0), and C(2, 3). Compute the auxiliary variables first:

  • a = 4 - 2 = 2, b = 0 - 3 = -3
  • c = 0 - 2 = -2, d = 0 - 3 = -3
  • e = 0*2 + 0*(-3) = 0, f = 4*(-2) + 0*(-3) = -8

Denominator: a*d - b*c = (2)(-3) - (-3)(-2) = -6 - 6 = -12

Hx = (0*(-3) - (-3)*(-8)) / (-12) = (0 - 24) / (-12) = 2

Hy = (2*(-8) - 0*(-2)) / (-12) = (-16 - 0) / (-12) = 4/3 approximately 1.333

The orthocenter of this triangle lies at the point (2, 4/3), confirmed by verifying that both altitudes from A and B pass through this coordinate.

Position by Triangle Type

The orthocenter's location shifts based on the angles of the triangle:

  • Acute triangle: All angles less than 90 degrees — orthocenter falls inside the triangle.
  • Right triangle: One angle equals exactly 90 degrees — orthocenter coincides with the vertex at the right angle.
  • Obtuse triangle: One angle exceeds 90 degrees — orthocenter falls outside the triangle, on the far side of the obtuse vertex.

The Euler Line Connection

One of the most celebrated results in classical geometry is that the orthocenter (H), centroid (G), and circumcenter (O) always lie on a single line called the Euler Line. Furthermore, the centroid divides segment HO in the ratio 2:1 measured from H. This property extends into non-Euclidean settings as explored in research archived at CSUSB ScholarWorks, underscoring the orthocenter's central role in triangle geometry at every level of mathematical sophistication.

Real-World Applications

The orthocenter appears in structural engineering when analyzing force vectors through triangular trusses, in computer graphics pipelines for barycentric shading and ray-triangle intersection, and in surveying calculations that use triangulation to determine positions. Standardized geometry curricula — including the New York State Next Generation Mathematics Learning Standards — require students to locate and construct orthocenters as a core high school geometry competency.

Reference

Frequently asked questions

What is the orthocenter of a triangle?
The orthocenter is the point where all three altitudes of a triangle meet. Each altitude is a perpendicular segment from one vertex to the line containing the opposite side. Every triangle has exactly one orthocenter, though its position — inside, on, or outside the triangle — depends on whether the triangle is acute, right, or obtuse respectively.
How does the orthocenter calculator find the answer?
The calculator solves a 2x2 linear system derived from two altitude equations. It computes six intermediate values (a, b, c, d, e, f) from the input coordinates, then applies Cramer's rule to find where the altitudes intersect. This approach reliably handles all triangle types and returns both the x-coordinate and y-coordinate of the orthocenter H as separate outputs.
Where is the orthocenter of a right triangle?
For a right triangle, the orthocenter coincides exactly with the vertex at the 90-degree angle. This is because the two legs of a right triangle are themselves altitudes — each leg is perpendicular to the other — and they already meet at the right-angle vertex. Using coordinates such as A(0,0), B(3,0), C(0,4) will return H = (0, 0), confirming this property.
Can the orthocenter be outside the triangle?
Yes. In an obtuse triangle — one where a single interior angle exceeds 90 degrees — the orthocenter always falls outside the triangular region, beyond the side opposite the obtuse angle. For example, a triangle with vertices A(0,0), B(10,0), C(9,1) is very obtuse near vertex A, and its orthocenter will have a large negative y-coordinate, placing it well below the base AB.
What is the Euler Line and how does the orthocenter relate to it?
The Euler Line is a straight line that passes through three of a triangle's classical centers: the orthocenter (H), the centroid (G), and the circumcenter (O). Leonhard Euler proved that these three points are always collinear and that the centroid lies exactly one-third of the way from the circumcenter to the orthocenter. This relationship holds for every non-equilateral triangle, making the orthocenter a critical anchor point in advanced triangle geometry.
What does it mean when the orthocenter formula denominator equals zero?
A denominator of zero — that is, a*d - b*c = 0 — means the three input points are collinear and do not form a triangle at all. No altitude intersection exists because the 'triangle' is degenerate. In this situation the orthocenter is mathematically undefined. Verify that the three vertices are not on the same line by checking that no two vertices share the same coordinates and that the points do not satisfy a single linear equation y = mx + b.