Orthocenter Calculator – Triangle Altitude Intersection
Find the orthocenter where triangle altitudes meet.
Table of Contents
How to Use
- Enter the x and y coordinates for vertex A
- Enter the x and y coordinates for vertex B
- Enter the x and y coordinates for vertex C
- Click calculate to find the orthocenter
- View the orthocenter coordinates and altitude equations
What is the Orthocenter?
The orthocenter is the point where all three altitudes of a triangle intersect. An altitude is a perpendicular line segment from a vertex to the opposite side (or its extension).
The orthocenter is one of the four main triangle centers, along with the centroid, circumcenter, and incenter.
Orthocenter Location by Triangle Type
The position of the orthocenter depends on the type of triangle:
- Acute triangle: Orthocenter is inside the triangle
- Right triangle: Orthocenter is at the vertex of the right angle
- Obtuse triangle: Orthocenter is outside the triangle
How to Calculate the Orthocenter
To find the orthocenter:
- Find the slope of each side of the triangle
- Calculate the perpendicular slope for each altitude
- Write the equation of each altitude line
- Find the intersection point of any two altitudes
Properties of the Orthocenter
Important properties of the orthocenter:
- All three altitudes always intersect at a single point
- The orthocenter, centroid, and circumcenter are collinear (Euler line)
- In an equilateral triangle, the orthocenter coincides with the centroid
- The reflection of the orthocenter over any side lies on the circumcircle
Frequently Asked Questions
- Why is the orthocenter outside for obtuse triangles?
- In an obtuse triangle, the altitude from the obtuse angle vertex falls outside the triangle because it must be perpendicular to the opposite side's extension. This causes all three altitudes to meet outside the triangle.
- What is the Euler line?
- The Euler line is a line that passes through the orthocenter, centroid, and circumcenter of any non-equilateral triangle. The centroid divides the segment from orthocenter to circumcenter in a 2:1 ratio.
- How is the orthocenter different from the centroid?
- The centroid is the intersection of medians (lines from vertices to midpoints of opposite sides), while the orthocenter is the intersection of altitudes (perpendicular lines from vertices to opposite sides). The centroid is always inside the triangle.
- What happens to the orthocenter in a right triangle?
- In a right triangle, the orthocenter is located exactly at the vertex where the right angle is formed. This is because the two legs of the triangle are already perpendicular to each other, serving as two of the altitudes.