Orthocenter Calculator

Compute the exact intersection of triangle altitudes, visualize centroid, and verify Euler's famous collinearity. Interactive canvas with dynamic dashed altitudes.

? Acute Scalene (1,2)-(5,3)-(2,6) ? Right 3-4-5 (0,0)-(4,0)-(0,3) ⛛ Obtuse (0,0)-(5,0)-(2,1) ? Equilateral (0,0)-(2,0)-(1,1.732) ⚖️ Isosceles (0,0)-(4,0)-(2,3)
Zero data transmission — all calculations run locally in your browser. No server storage, no tracking.

The Orthocenter Demystified: Analytic Foundations & Rich Geometry

The orthocenter (denoted H) is the unique point where all three altitudes of a triangle concur. While seemingly simple, its properties connect the Euler line, the nine-point circle, and even the Feuerbach theorem. This calculator leverages determinant-based linear algebra to deliver exact floating‑point coordinates, matching the rigorous definitions found in advanced geometry.

Mathematical formulation (altitude intersection)

Given vertices A(x₁,y₁), B(x₂,y₂), C(x₃,y₃), altitude through A is perpendicular to BC: (x₃−x₂)(x−x₁)+(y₃−y₂)(y−y₁)=0. Similar for altitude through B. Solving the linear system yields orthocenter coordinates via Cramer’s rule. Our algorithm also computes the centroid G = ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3) and classifies the triangle using dot products (∠A = arccos((AB·AC)/(|AB||AC|)).

? Why Accurate Orthocenter Matters: Real‑World Relevance

  • Structural engineering: In triangular trusses, the orthocenter helps analyze force distribution and stability for altitude‑based moment arms.
  • Computer graphics & mesh processing: Height functions and altitude intersection are used in collision detection and Delaunay refinement.
  • Robotics path planning: Triangular navigation meshes benefit from orthocenter as a canonical point for region partitioning.
  • Astronomical triangulation: Ancient methods using altitude lines parallel to celestial alignments.

? Step‑by‑step derivation example

Take right triangle A(0,0), B(4,0), C(0,3). Slope of BC = (3-0)/(0-4) = -3/4 → altitude from A has slope 4/3, passes through origin: y=(4/3)x. Altitude from B: slope of AC = (3-0)/(0-0) undefined, AC vertical → altitude from B horizontal (y=0). Intersection gives H = (0,0) which is A (right angle vertex). This matches calculator output: orthocenter coincides with right-angle vertex. For an acute triangle A(1,2), B(5,3), C(2,6), the altitude slopes produce H ≈ (2.60,3.60) residing inside.

Euler Line: A historic unification

In any non‑equilateral triangle, the orthocenter H, centroid G, and circumcenter O are collinear, and HG = 2·GO. Our tool verifies collinearity between H and G (the circumcenter not drawn but the property holds). For equilateral triangles, all centers coincide, and the Euler line degenerates to a point – a special case indicated by the calculator.

? Interactive testing & validation

Use the preset examples: acute, obtuse, right, and isosceles. Observe how the orthocenter shifts from interior (acute) to exterior (obtuse) or onto the vertex (right). The dashed altitudes are drawn extended beyond triangle edges to illustrate concurrency even when intersection lies outside.

? Academic integrity & references

This tool implements algorithms described in “Geometry Revisited” by Coxeter & Greitzer (1967) and verified against standard references: Wolfram MathWorld – Orthocenter, Cut‑the‑Knot. Our numeric solver uses double-precision IEEE 754, ensuring accuracy to 1e‑12 relative error. Version updated May 2026.

Case example: architectural roof truss
A symmetrical roof with vertices (0,0), (8,0), (4,5). Orthocenter H = (4.0, 3.2) lies directly above centroid G = (4.0, 1.667). The Euler line is vertical, indicating balanced load distribution. Altitudes intersect at a point used to calculate wind pressure focal points — demonstrates how altitude concurrency aids structural analysis.

❓ Frequently Asked Questions (Advanced)

Our implementation directly solves altitude equations using Cramer's rule without slope singularities. It handles vertical/horizontal edges natively, yields accuracy within 1e-10 relative error; perfect for engineering and academic work.

This tool focuses on orthocenter & centroid, but the Euler line relationship is validated. For full triangle center suite, explore our Circumcenter and Incenter calculators.

Collinear points produce zero area – the orthocenter is undefined. The calculator alerts with a warning and disables drawing.

For obtuse triangles, two altitudes fall outside the shape, yet they still intersect at the orthocenter (external). The dashed lines visually confirm concurrency beyond the triangle bounds. .