1. Definition
Euler line: the centroid G, circumcenter O and orthocenter H of any triangle are collinear, and G divides OH with HG:GO=2:1 (G is closer to O).
2. Symbols
| Symbol | Meaning |
|---|
| G | centroid (average of vertices) |
| O | circumcenter |
| H | orthocenter |
| HG:GO=2:1 | division ratio |
3. How it works
- Centroid G=(A+B+C)/3;
- Circumcenter O is the intersection of perpendicular bisectors;
- Orthocenter H=A+B+C−2O;
- They are collinear with HG:GO=2:1.
4. Steps
- Enter three vertex coordinates;
- Click Calculate;
- Read G, O, H;
- Verify collinearity and the 2:1 ratio.
5. Example
Example: triangle (0,0)(4,0)(0,3). G≈(1.333,1), O=(2,1.5), H=(0,0); they are collinear with HG:GO=2:1.
6. Pitfalls
The three points must not be collinear;
In an equilateral triangle the centers coincide and the line degenerates to a point;
G always lies between O and H.