Orthocentre Calculator

Understanding the Orthocentre of a Triangle

In geometry, the orthocentre is one of the fundamental points associated with a triangle. It's a concept that has fascinated mathematicians for centuries, playing a crucial role in understanding the intricate relationships within a triangle's structure.

What is an Orthocentre?

The orthocentre is defined as the point where the three altitudes of a triangle intersect. An altitude of a triangle is a line segment from a vertex perpendicular to the opposite side (or to the line containing the opposite side). Every triangle has exactly three altitudes, and remarkably, these three lines always meet at a single point – the orthocentre.

The position of the orthocentre varies depending on the type of triangle:

• Acute Triangle: For an acute triangle (where all angles are less than 90 degrees), the orthocentre lies inside the triangle.
• Right Triangle: In a right-angled triangle, the orthocentre coincides with the vertex where the right angle is located. This is because the two legs of the right triangle are already altitudes.
• Obtuse Triangle: For an obtuse triangle (where one angle is greater than 90 degrees), the orthocentre lies outside the triangle. This happens because the altitudes from the acute vertices must extend beyond the triangle's sides to meet the lines containing the opposite sides perpendicularly.

How to Calculate the Orthocentre Using Coordinates

Calculating the orthocentre of a triangle given its vertices in a Cartesian coordinate system involves finding the equations of at least two altitudes and then determining their intersection point. Our calculator above automates this process, but understanding the steps can deepen your appreciation for this geometric concept.

The Step-by-Step Method:

1. Define the Vertices: Let the three vertices of the triangle be A(x₁, y₁), B(x₂, y₂), and C(x₃, y₃).
2. Determine the Slope of a Side: Choose two sides of the triangle, for example, BC and AC.
   • Slope of BC (m_BC) = (y₃ - y₂) / (x₃ - x₂)
   • Slope of AC (m_AC) = (y₃ - y₁) / (x₃ - x₁)
   Special cases: If a side is vertical (x-coordinates are the same), its slope is undefined. If a side is horizontal (y-coordinates are the same), its slope is zero.
3. Find the Slope of the Altitude: An altitude is perpendicular to its corresponding side. The product of the slopes of two perpendicular lines is -1.
   • Slope of altitude from A to BC (m_AD) = -1 / m_BC
   • Slope of altitude from B to AC (m_BE) = -1 / m_AC
   Special cases: If a side is vertical, the altitude perpendicular to it is horizontal (slope = 0). If a side is horizontal, the altitude perpendicular to it is vertical (undefined slope).
4. Write the Equation of Two Altitudes: Use the point-slope form of a linear equation (y - y₀ = m(x - x₀)).
   • Equation of altitude AD: Passes through A(x₁, y₁) with slope m_AD.
   • Equation of altitude BE: Passes through B(x₂, y₂) with slope m_BE.
   For vertical altitudes, the equation is simply x = x-coordinate of the vertex. For horizontal altitudes, y = y-coordinate of the vertex.
5. Solve the System of Equations: The orthocentre is the point (x, y) that satisfies both altitude equations. Solve the two linear equations simultaneously to find the coordinates of the orthocentre.

Geometric Significance and Related Concepts

The orthocentre is one of the four classical "centres" of a triangle, alongside the centroid, circumcentre, and incentre. These points are often studied together due to their interconnectedness.

• Euler Line: For any non-equilateral triangle, the orthocentre (H), the centroid (G), and the circumcentre (O) are collinear, meaning they all lie on a single straight line called the Euler line. The centroid G always lies between H and O, and the distance HG is twice the distance GO.
• Nine-Point Circle: The orthocentre also plays a role in defining the nine-point circle, which passes through nine significant points of the triangle, including the midpoints of the sides, the feet of the altitudes, and the midpoints of the segments connecting the vertices to the orthocentre.
• Orthic Triangle: The triangle formed by joining the feet of the altitudes is known as the orthic triangle. The orthocentre of the original triangle is the incentre of its orthic triangle.

While the orthocentre might not have as many direct real-world applications as, say, the centroid (center of mass), its theoretical importance in geometry and its connections to other fundamental triangle properties make it a cornerstone of advanced geometric study.