Euler Line Calculator
Understanding the Euler Line
What is the Euler Line?
The Euler line is a remarkable straight line that contains several important points of a triangle:
- Orthocenter (H)
- Centroid (G)
- Circumcenter (O)
- Nine-point center (N)
These points always lie on the same line, except in equilateral triangles where they coincide.
Key Properties
- HG:GO = 2:1
- N is midpoint of HO
- G divides HO in ratio 2:1
- Exists in all triangles
- Parallel to longest side in obtuse triangles
Triangle Centers
Orthocenter (H)
Intersection of altitudes
Centroid (G)
Intersection of medians
Circumcenter (O)
Center of circumscribed circle
Nine-point center (N)
Center of nine-point circle
Special Cases
| Triangle Type | Euler Line Property |
|---|---|
| Equilateral | All centers coincide |
| Right | H at right angle vertex |
| Obtuse | H outside triangle |
| Acute | H inside triangle |
Advanced Properties
Distance Relationships
- OH = 3R cos A cos B cos C
- ON = R/2
- HN = 2R cos A cos B cos C
Circle Properties
- Nine-point circle radius = R/2
- Euler circle contains feet of altitudes
- Contains midpoints of sides
Applications and Extensions
Geometry
Triangle center relationships
Engineering
Structural analysis
Physics
Center of mass calculations
Architecture
Design and symmetry