Orthocentre Calculator

Category: Geometry
- October 26, 2025
| |

Calculate and visualise the orthocentre of a triangle. The orthocentre is the point where all three altitudes of a triangle intersect. This calculator helps understand the geometric properties of triangles and their orthocentre.

Triangle Coordinates

Display Options

About the Orthocentre

The orthocentre is the point where all three altitudes of a triangle intersect. An altitude of a triangle is a line segment from a vertex perpendicular to the opposite side.

Properties of the Orthocentre
  • Location: The orthocentre can be inside, outside, or on the triangle, depending on the triangle type.
  • For acute triangles: The orthocentre is inside the triangle.
  • For right triangles: The orthocentre is at the vertex with the right angle.
  • For obtuse triangles: The orthocentre is outside the triangle, opposite to the obtuse angle.
  • Equilateral triangles: The orthocentre coincides with the incentre, centroid, and circumcentre.
Mathematical Calculation

To find the orthocentre:

  1. Find the equations of two altitudes of the triangle
  2. Solve these equations to find their intersection point
  3. The third altitude will also pass through this point

Alternatively, we can use coordinate geometry formulas to directly calculate the orthocentre.

Applications

The concept of the orthocentre is used in:

  • Geometric design and construction
  • Engineering and physics for calculating centres of structures
  • Computer graphics for shape manipulation
  • Navigation and triangulation methods

Introduction to the Orthocenter Calculator

The Orthocenter Calculator is a handy tool that helps you find the orthocenter of a triangle. The orthocenter is the point where the three altitudes of the triangle intersect. Knowing this point is useful in Geometry and helps you understand the properties of triangles better.

Understanding Triangle Coordinates

To use the calculator, you need to input the coordinates of the triangle’s vertices. The three points are labelled A, B, and C. Each point has an x and a y coordinate. This information is crucial as the calculator uses these points to determine the orthocenter's location accurately.

Why Use the Orthocenter Calculator?

This calculator offers several benefits, including:

  • Quick and accurate calculation of the orthocenter.
  • Visual representation of the triangle and its altitudes.
  • Detailed results, including the lengths of the altitudes from each vertex.

The Role of Altitudes in Triangles

In geometry, an altitude is a line from a vertex that is perpendicular to the opposite side. Each triangle has three altitudes. These are significant because the orthocenter is formed where all three altitudes intersect. This intersection point can provide insight into various triangle properties.

How the Calculator Works

The Orthocenter Calculator takes your input and performs calculations to find the orthocenter. It calculates the equations of two altitudes and determines their intersection point. The calculator also provides the lengths of the altitudes, which helps you understand the triangle's dimensions better.

Exploring the Results

Once you input the triangle coordinates, the calculator displays the orthocenter's coordinates. It also shows the lengths of the altitudes from each vertex. These results are not only numbers; they help you realise how the triangle behaves and its structural properties.

Visualising Triangles and Altitudes

The calculator provides a visual representation of the triangle and its altitudes. This feature allows you to see how the orthocenter relates to the triangle's shape. A clear diagram can enhance your understanding of triangle geometry and the significance of the orthocenter.

Applications of the Orthocenter

The concept of the orthocenter has various applications, including:

  • Geometric design in architecture.
  • Physics for understanding structures and forces.
  • Computer graphics for modelling shapes.
  • Navigation techniques using triangulation.