Orthocenter Finder Calculator
Understanding the Orthocenter
What is an Orthocenter?
The orthocenter is the point where the three altitudes of a triangle intersect. An altitude is a line segment from a vertex perpendicular to the opposite side (or its extension).
- Unique point for every triangle
- Intersection of three altitudes
- Key center in triangle geometry
- Related to other triangle centers
Properties of Altitudes
- Always perpendicular to base
- Shortest distance to opposite side
- Equal in equilateral triangles
- One coincides with leg in right triangles
- Forms similar triangles
Triangle Classification by Orthocenter
Acute Triangle
Orthocenter inside
Right Triangle
Orthocenter at vertex
Obtuse Triangle
Orthocenter outside
Equilateral
Coincides with centroid
Mathematical Properties
| Property | Description |
|---|---|
| Distance Formula | d = |ax + by + c| / √(a² + b²) |
| Area Relation | A = (1/2) × base × height |
| Perpendicular Lines | m₁ × m₂ = -1 |
| Altitude Length | h = 2A/b |
Advanced Relationships
Euler's Theorem
- OH = 3R cos A cos B cos C
- H lies on Euler line
- Related to circumcenter
Distance Properties
- Product of distances
- Relation to circumradius
- Pedal triangle area
Real-World Applications
Engineering
Structural stability analysis
Architecture
Building design optimization
Physics
Center of gravity calculations
Navigation
Triangulation methods