What Is the Orthocenter of a Triangle?
The orthocenter is one of the four main triangle centers (along with the centroid, circumcenter, and incenter). It is the point where all three altitudes of a triangle intersect. An altitude is a line segment drawn from a vertex perpendicular to the opposite side (or its extension).
How to Find the Orthocenter
Step 1: Find Slopes
Calculate the slope of each side of the triangle.
m = (y2 - y1) / (x2 - x1)
Step 2: Perpendicular Slopes
The altitude from a vertex is perpendicular to the opposite side.
m_perp = -1 / m_side
Step 3: Find Intersection
Write altitude equations and solve for the intersection point.
y - y0 = m_perp(x - x0)
Orthocenter Position
- Acute triangle: The orthocenter lies inside the triangle.
- Right triangle: The orthocenter is at the vertex of the right angle.
- Obtuse triangle: The orthocenter lies outside the triangle.
Euler Line
A remarkable property of the orthocenter is that it lies on the Euler line along with the centroid and circumcenter. The centroid divides the segment from the orthocenter to the circumcenter in a 2:1 ratio. This relationship holds for all non-equilateral triangles.
Properties of the Orthocenter
- The reflection of the orthocenter over the midpoint of any side lies on the circumcircle.
- In an equilateral triangle, the orthocenter, centroid, circumcenter, and incenter all coincide.
- The orthocenter, along with the three vertices, forms an orthocentric system where each point is the orthocenter of the triangle formed by the other three.