Orthocenter of Triangle, Altitude Calculation

Calculate the orthocenter of a triangle with the entered values of coordinates.

Calculator

Please enter the coordinates of a traingle:

xy


The orthocenter of a triangle is described as a point where the altitudes of triangle meet and altitude of a triangle is a line which passes through a vertex of the triangle and is perpendicular to the opposite side, therefore three altitudes possible, one from each vertex. We can say that all three altitudes always intersect at the same point is called orthocenter of the triangle.

orthocenter of a triangle

What Is the Orthocenter of a Triangle?

The orthocenter is the common intersection point of the three altitudes of a triangle. An altitude is a line segment drawn from a vertex perpendicular to the opposite side. This calculator determines the exact coordinates of the orthocenter using the three vertex coordinates entered by the user.

The location of the orthocenter depends on the type of triangle:

  • Inside an acute triangle
  • At the right-angle vertex of a right triangle
  • Outside an obtuse triangle

This tool automatically computes altitude equations, perpendicular slopes, and the orthocenter coordinates while displaying each step of the calculation.

In the case triangle is obtuse (the triangle where one of the internal angles is greater than 90 degrees) than orthocenter will be outside. therefore we can say that orthocenter is not always inside the triangle.
There is 4 steps to calculate the orthocenter of any triangle as described below.
Step 1: Calculate the slope of the sides of the triangle using following formula: $$ \text{Slope of a line} = \frac{y\,_{2}-y\,_{1}}{x\,_{2}-x\,_{1}}$$ Step 2: Calculate the perpendicular slope of the sides using following formula: $$ \text{Perpendicular slope of the line } = \frac{-1}{ \text{Slope of a line} }$$ Step 3: Calculate the equation for any two altitudes with their respective coordinates using following formula: $$ y - y\,_{1} = m(x-x\,_{1})$$ Step 4: Solving altitude equations (any two altitude equation of Step 3)

What is the Orthocenter of a Triangle?

The orthocenter is the point where the three altitudes of a triangle intersect. An altitude is a line segment drawn from a vertex perpendicular to the opposite side. Every triangle has exactly one orthocenter.

Where is the Orthocenter Located?

  • Acute Triangle → Inside the triangle
  • Right Triangle → At the right-angle vertex
  • Obtuse Triangle → Outside the triangle

Orthocenter Formula

The orthocenter is obtained by finding the intersection point of any two altitudes. The calculator automatically determines the altitude equations and solves them simultaneously.

Applications of Orthocenter

  • Coordinate Geometry
  • Engineering Design
  • Surveying
  • Architecture
  • Computer Graphics
  • Mathematics Education

Frequently Asked Questions

How do you find the orthocenter of a triangle?

Find the equations of any two altitudes and calculate their intersection point.

Can the orthocenter lie outside a triangle?

Yes. In an obtuse triangle the orthocenter lies outside the triangle.

Where is the orthocenter of a right triangle?

The orthocenter is located at the vertex containing the 90° angle.

Do all triangles have an orthocenter?

Yes. Every non-degenerate triangle has exactly one orthocenter.

What is the difference between orthocenter and centroid?

The orthocenter is formed by altitudes, whereas the centroid is formed by medians.