Orthocentre of a Triangle

In Euclidean geometry, the orthocentre is one of the most significant points associated with a triangle. It represents the point of intersection of all three altitudes of a triangle. An altitude is a line segment drawn from a vertex perpendicular to the opposite side (or its extension).

The orthocentre, denoted by H, holds special properties that make it essential in both theoretical geometry and practical applications. Depending on the triangle's shape, the orthocentre can be located inside, outside, or even on the triangle.

Orthocentre Triangle Definition

For a triangle with vertices A, B, and C, the orthocentre H is the point of intersection of:

• The altitude from A to side BC
• The altitude from B to side AC
• The altitude from C to side AB

Mathematical notation:

H = h_A ∩ h_B ∩ h_C

Key Properties of the Orthocentre

• The orthocentre is equidistant from the feet of the altitudes.
• If H is the orthocentre of triangle ABC, then A is the orthocentre of triangle HBC.
• The orthocentre lies on the Euler line along with the centroid, circumcenter, and nine-point circle center.
• The distance from the orthocentre to any side is inversely proportional to the distance from the opposite vertex.

Orthocentre History

The concept of the orthocentre dates back to ancient Greek mathematics. While Euclid's Elements discusses perpendicular lines, the formal recognition of the orthocentre came later in geometry's development.

Calculating the Orthocentre

Method 1: Using Coordinate Geometry

Given vertices A(x₁, y₁), B(x₂, y₂), C(x₃, y₃), the orthocentre H(x, y) can be found as follows:

1. Equation of side BC:
   (y₃ - y₂)(x - x₂) + (x₂ - x₃)(y - y₂) = 0
2. Equation of the altitude from A:
   Slope of BC = (y₃ - y₂)/(x₃ - x₂)
   Slope of altitude from A = -1 / (Slope of BC)
   Equation: (y - y₁) = m(x - x₁)
3. Similarly, find altitudes from B and C.
4. Solve any two altitude equations simultaneously to find (x, y).

Example 1: Finding the Orthocentre

Given triangle A(0, 0), B(6, 0), and C(3, 4):

• Slope of BC = (4 - 0) / (3 - 6) = -4/3
• Perpendicular slope (altitude from A) = 3/4
• Equation of altitude from A: y = (3/4)x
• Equation of altitude from B: y = -3/4(x - 6)
• Altitude from C: x = 3

Solution:
Substituting x=3 into y=(3/4)x:
y = 9/4 = 2.25
Thus, H(3, 2.25).

Method 2: Using Barycentric Coordinates

For angles α, β, γ at vertices A, B, C respectively:

H = (−tan α : −tan β : −tan γ)

Alternatively, using side lengths a, b, and c:

H = (cos A × cot A : cos B × cot B : cos C × cot C)

Orthocentre in Different Types of Triangles

Important Theorems and Relations

Euler Line Theorem

The centroid G, circumcenter O, and orthocentre H are collinear:

G = (2O + H)/3

Nine-Point Circle Theorem

The nine-point circle passes through:

• Midpoints of sides
• Feet of altitudes
• Midpoints between H and each vertex

Center N of the circle lies at:

N = (O + H)/2

Applications of the Orthocentre

Engineering and Physics

• Structural analysis of trusses
• Center of mass calculations
• Electromagnetic field modeling

Computer Graphics

• Mesh generation for simulations
• 3D triangulation in computer vision
• Spatial analysis in GIS systems

Visual Representation:

Geometric Construction of the Orthocentre

Steps:

1. Draw the triangle.
2. Construct the perpendiculars from each vertex to the opposite side.
3. The point of intersection is the orthocentre.

Construction Software:

• GeoGebra
• Cabri Geometry
• Cinderella

The orthocentre is a remarkable point in triangle geometry, deeply connected to many important theorems and applications. Mastering the orthocentre strengthens understanding of triangle properties, Euclidean structures, and modern computational applications.