🔺 Orthocenter Calculator
Calculate the orthocenter of a triangle
Triangle Vertices
How to Use This Calculator
Enter Triangle Vertices
Input the x and y coordinates for three vertices (A, B, C) of your triangle. Make sure the points are not collinear (do not lie on a straight line).
Click Calculate
Press the "Calculate Orthocenter" button to find the point where all three altitudes intersect.
Review Results
View the orthocenter coordinates displayed below.
Formula
Orthocenter = Intersection point of three altitudes
Altitude: Perpendicular line from vertex to opposite side
Method:
- Find the slopes of two sides of the triangle
- Calculate the slopes of the altitudes (perpendicular to these sides)
- Find the intersection point of two altitudes
- This intersection point is the orthocenter
Special Cases:
- Right Triangle: Orthocenter is at the right angle vertex
- Acute Triangle: Orthocenter is inside the triangle
- Obtuse Triangle: Orthocenter is outside the triangle
About Orthocenter Calculator
The Orthocenter Calculator finds the orthocenter of a triangle - the point where all three altitudes intersect. An altitude is a line segment drawn from a vertex perpendicular to the opposite side.
When to Use This Calculator
- Geometry: Find the orthocenter for triangle geometry problems
- Education: Learn about triangle centers and their properties
- Engineering: Calculate triangle centers for design and analysis
- Computer Graphics: Locate triangle centers for rendering and transformations
- Surveying: Find special points in triangular land plots
- Architecture: Design triangular structures and find balance points
Why Use Our Calculator?
- ✅ Instant Results: Get orthocenter coordinates immediately
- ✅ Handles All Cases: Works for acute, right, and obtuse triangles
- ✅ Accurate Calculations: Precise mathematical computations
- ✅ Easy Input: Simply enter three vertex coordinates
- ✅ 100% Accurate: Precise geometric calculations
- ✅ Completely Free: No registration required
Understanding Orthocenter
The orthocenter is one of four notable triangle centers (along with centroid, circumcenter, and incenter). Key properties:
- Location: Position depends on triangle type: inside (acute), at vertex (right), or outside (obtuse)
- Altitudes: Three altitudes always meet at the orthocenter
- Euler Line: In most triangles, orthocenter, centroid, and circumcenter lie on the same line (Euler line)
- Equilateral Triangle: In equilateral triangles, all four centers coincide
- Relationship: Orthocenter is the reflection of circumcenter about the centroid
Real-World Applications
Engineering: Triangle centers are important in structural analysis. The orthocenter can represent optimal load distribution points in triangular trusses.
Navigation: In triangulation methods, triangle centers help determine positions and distances.
Design: Triangle centers guide placement of elements in triangular designs, ensuring balance and visual appeal.
Frequently Asked Questions
What is an orthocenter?
The orthocenter is the point where all three altitudes of a triangle intersect. An altitude is a line drawn from a vertex perpendicular to the opposite side.
Can the orthocenter be outside the triangle?
Yes! In an obtuse triangle, the orthocenter lies outside the triangle. In acute triangles, it's inside. In right triangles, it's at the right angle vertex.
What's the difference between orthocenter, centroid, and circumcenter?
Orthocenter: intersection of altitudes. Centroid: intersection of medians (balance point). Circumcenter: intersection of perpendicular bisectors (center of circumscribed circle).
Do all triangles have an orthocenter?
Yes, all triangles have an orthocenter. However, if the three vertices are collinear (form a line), they don't form a triangle, so there's no orthocenter.
Where is the orthocenter in an equilateral triangle?
In an equilateral triangle, the orthocenter, centroid, circumcenter, and incenter all coincide at the same point (the center of the triangle).
What is the Euler line?
The Euler line is a line passing through the orthocenter, centroid, and circumcenter. In most triangles (except equilateral), these three points are collinear and lie on the Euler line.
How do I find the orthocenter manually?
To find it manually: 1) Draw altitudes from each vertex to the opposite side, 2) Find where any two altitudes intersect - that's the orthocenter. All three altitudes meet at this point.