Triangle Orthocenter Calculator

Welcome to the Triangle Orthocenter Calculator — an interactive tool that finds the orthocenter (intersection of the three altitudes) of any triangle from its vertex coordinates, with a live diagram showing altitudes, the Euler line, step-by-step solutions, and full triangle analysis. Whether you are a geometry student, an engineer working with coordinate geometry, or a math enthusiast, this calculator makes orthocenter computation instant and visual.

What is the Orthocenter of a Triangle?

The orthocenter of a triangle is the point where all three altitudes intersect. An altitude is a line segment drawn from a vertex perpendicular to the opposite side (or its extension). The orthocenter is one of the four classical triangle centers, alongside the centroid, circumcenter, and incenter.

Orthocenter Formula

For a triangle with vertices A(x₁, y₁), B(x₂, y₂), and C(x₃, y₃), the orthocenter H(Hx, Hy) is found by solving the system of perpendicularity equations:

This yields a linear system of two equations in two unknowns (Hx and Hy), solvable via Cramer's rule or substitution.

Where Does the Orthocenter Lie?

Unlike the centroid (which always lies inside), the orthocenter's position depends on the triangle type:

• Acute triangle: The orthocenter lies inside the triangle.
• Right triangle: The orthocenter coincides with the vertex at the right angle.
• Obtuse triangle: The orthocenter lies outside the triangle, beyond the side opposite the obtuse angle.

The Euler Line

For any non-equilateral triangle, three important centers are collinear on the Euler line:

• Circumcenter (O) — center of the circumscribed circle
• Centroid (G) — center of mass (median intersection)
• Orthocenter (H) — altitude intersection

The centroid divides the segment OH in a 1:2 ratio from O, meaning OG:GH = 1:2. This powerful relationship connects three seemingly unrelated triangle properties.

How to Use This Calculator

1. Enter coordinates: Input x and y values for vertices A, B, and C. Negative numbers and decimals are supported.
2. Choose precision: Select your preferred number of decimal places (2 to 10).
3. Click Calculate: The orthocenter H = (Hx, Hy) is displayed with a full breakdown and interactive diagram.
4. Explore the diagram: See the triangle, its three color-coded altitudes with right-angle markers, altitude feet, the animated orthocenter, and the Euler line connecting H, G, and O.

Orthocenter vs Other Triangle Centers

Properties of the Orthocenter

• Altitude concurrency: The three altitudes of any triangle always meet at a single point — the orthocenter. This is a consequence of Ceva's theorem.
• Euler line: H, G, and O are collinear (except for equilateral triangles where they coincide).
• Reflection property: Reflecting the orthocenter over the midpoint of any side places it on the circumscribed circle.
• Orthocentric system: If H is the orthocenter of triangle ABC, then each vertex is the orthocenter of the triangle formed by the other two vertices and H.
• Distance relation: The sum of distances from the orthocenter to the vertices equals twice the sum from the circumcenter to the vertices.

Frequently Asked Questions

What is the orthocenter of a triangle?

The orthocenter is the point where the three altitudes of a triangle intersect. An altitude is a perpendicular line segment from a vertex to the opposite side. It is one of the four classical triangle centers and lies on the Euler line.

How do you find the orthocenter of a triangle with coordinates?

Set up two perpendicularity equations using the dot product condition: AH·BC = 0 and BH·AC = 0. This gives a 2×2 linear system that you solve for the orthocenter coordinates (Hx, Hy) using Cramer's rule. This calculator performs all these steps automatically.

Is the orthocenter always inside the triangle?

No. The orthocenter is inside only for acute triangles. For right triangles, it sits at the right-angle vertex. For obtuse triangles, it lies outside the triangle. This is what makes the orthocenter unique among triangle centers.

What is the Euler line?

The Euler line is a straight line through three triangle centers: the circumcenter (O), centroid (G), and orthocenter (H). The centroid divides segment OH in a 1:2 ratio from O. For equilateral triangles, all three coincide, so no unique line exists.

What is the difference between the orthocenter and the centroid?

The orthocenter is where the three altitudes (perpendicular to opposite sides) meet, while the centroid is where the three medians (to midpoints of opposite sides) meet. The centroid is always inside the triangle and is its center of mass. The orthocenter can be outside for obtuse triangles.

Additional Resources

• Altitude (Triangle) - Wikipedia
• Euler Line - Wikipedia
• Triangle Center - Wikipedia