Orthocenter Calculator

Determine the intersection point of a triangle's altitudes.

Provide the coordinates for the three vertices of a triangle to compute the location of its orthocenter.

Practical Examples

Explore different triangle types and see how the orthocenter changes. Click on any example to load it into the calculator.

Acute Triangle

acute

An acute triangle where the orthocenter lies inside the triangle.

A: (2, 3)

B: (8, 1)

C: (5, 7)

Obtuse Triangle

obtuse

An obtuse triangle where the orthocenter lies outside the triangle.

A: (2, 2)

B: (4, 6)

C: (9, 1)

Right-Angled Triangle

right

A right-angled triangle where the orthocenter coincides with the right-angle vertex.

A: (0, 0)

B: (5, 0)

C: (0, 3)

General Triangle

general

A standard triangle to demonstrate a general use case.

A: (-2, 1)

B: (3, 2)

C: (1, 5)

Other Titles
Understanding the Orthocenter: A Comprehensive Guide
A deep dive into one of the fundamental centers of a triangle, exploring its calculation, properties, and applications.

What is the Orthocenter?

  • Defining the Orthocenter
  • The Role of Altitudes
  • Orthocenter Location by Triangle Type
The orthocenter is a point of concurrency in a triangle, meaning it's a point where three special lines intersect. Specifically, it is the intersection point of the triangle's three altitudes.
Understanding Altitudes
An altitude of a triangle is a line segment drawn from a vertex perpendicular to the opposite side (or the extension of the opposite side). Since every triangle has three vertices, it also has three altitudes. The fact that these three altitudes always meet at a single point—the orthocenter—is a fundamental property of triangles.
How the Orthocenter's Position Varies
The location of the orthocenter provides clues about the triangle's angles:
Acute Triangle: In a triangle where all angles are less than 90°, the orthocenter lies inside the triangle.
Right-Angled Triangle: In a triangle with one 90° angle, the orthocenter coincides with the vertex of the right angle.
Obtuse Triangle: In a triangle with one angle greater than 90°, the orthocenter lies outside the triangle.

Key Concepts

  • An altitude is always perpendicular to the side it intersects.
  • The three altitudes of any triangle are always concurrent.

Step-by-Step Guide to Using the Orthocenter Calculator

  • Inputting Vertex Coordinates
  • Interpreting the Results
  • Using the Examples
Our calculator simplifies finding the orthocenter. Here's how to use it effectively:
1. Enter the Vertex Coordinates
The calculator requires the Cartesian coordinates (x, y) for each of the three vertices of the triangle, labeled A, B, and C. Input the corresponding x and y values into the designated fields.
2. Calculate and Review
Click the 'Calculate' button. The tool will instantly compute the result, displaying the coordinates of the orthocenter (H), the type of triangle (acute, obtuse, or right), and its area. If the points do not form a valid triangle (i.e., they are collinear), an error message will appear.
3. Reset or Load an Example
Use the 'Reset' button to clear all inputs for a new calculation. You can also click on any of the provided examples to automatically populate the input fields with pre-defined values for different triangle types.

Usage Notes

  • Ensure all six input fields (x and y for each of the three vertices) are filled.
  • The calculator handles both positive and negative coordinates, as well as decimals.

Real-World Applications of the Orthocenter

  • Engineering and Physics
  • Computer Graphics
  • Geometric Problem Solving
While more of a pure geometric concept, the principles behind the orthocenter appear in various fields.
Structural Engineering
The concept of perpendicularity is fundamental in engineering. The orthocenter relates to the study of forces and stability in triangular structures, such as trusses, where understanding force vectors (which can be modeled as altitudes) is crucial.
Computer Graphics and Robotics
In computer-aided design (CAD) and robotics, geometric constructions are essential. Calculating points like the orthocenter is necessary for defining object properties, determining paths, and handling geometric transformations in 2D and 3D space.

Application Fields

  • Analyzing stress points in a triangular mechanical bracket.
  • Programming the movement of a robotic arm within a triangular workspace.

Common Misconceptions and Correct Methods

  • Orthocenter vs. Centroid vs. Circumcenter
  • Handling Vertical and Horizontal Lines
  • The 'Collinear' Edge Case
It's easy to confuse the different triangle centers. Let's clarify some common points of confusion.
Distinguishing Triangle Centers
Orthocenter: Intersection of altitudes (perpendicular lines from vertex to opposite side).
Centroid: Intersection of medians (lines from vertex to the midpoint of the opposite side). This is the triangle's center of mass.
Circumcenter: Intersection of perpendicular bisectors of the sides. It is the center of the circle that passes through all three vertices.
Dealing with Special Cases
The calculation must correctly handle cases where a triangle side is perfectly horizontal or vertical. If a side is horizontal, its altitude is a vertical line (with an undefined slope). If a side is vertical, its altitude is a horizontal line (with a slope of zero). Our calculator correctly manages these scenarios to provide an accurate result.

Important Distinctions

  • The orthocenter, centroid, and circumcenter are only the same point for an equilateral triangle.
  • For a right triangle, the orthocenter is on a vertex, while the circumcenter is at the midpoint of the hypotenuse.

Mathematical Derivation and Formulas

  • Finding the Slope of a Line
  • The Perpendicular Slope Rule
  • Solving a System of Linear Equations
The orthocenter is found by determining the intersection of two altitude lines. Here's the mathematical process:
1. Find Slopes of Triangle Sides
Given two vertices, A(x₁, y₁) and B(x₂, y₂), the slope of the side AB is m_AB = (y₂ - y₁) / (x₂ - x₁).
2. Determine Slopes of Altitudes
The altitude from vertex C to side AB is perpendicular to AB. Its slope, malt, is the negative reciprocal of mAB. So, malt = -1 / mAB.
3. Formulate Line Equations
Using the point-slope form, y - y₀ = m(x - x₀), we can write the equations for two different altitudes. For example, the altitude from C(x₃, y₃) to AB has the equation: y - y₃ = (-1 / m_AB) * (x - x₃).
4. Solve the System of Equations
By creating equations for two altitudes and solving them simultaneously, we find the (x, y) coordinate pair where they intersect. This point is the orthocenter.

Formulaic Steps

  • If a side's slope is 2, the altitude's slope is -1/2.
  • If a side is horizontal (slope = 0), the altitude is vertical (undefined slope, equation is x = constant).