Orthocenter Calculator
Type the coordinates of vertices A, B, and C — negatives and decimals fine — and get the orthocenter to 3 decimals, plus the centroid and circumcenter so you can see the whole Euler line.
Example: with Vertex A — x₁ 1 · Vertex A — y₁ 1 · Vertex B — x₂ 7 · Vertex B — y₂ 1 · Vertex C — x₃ 3 → Orthocenter (x, y): (3, 3).
- Triangle typeAcute — the orthocenter lies inside the triangle.
- Centroid (bonus)(3.667, 2.333)
- Circumcenter (bonus)(4, 2)
Computed by the calculator below using its default values. Change any input to see your own numbers.
Altitudes are concurrent: solving any two of them locates the orthocenter. It shares the Euler line with the centroid and circumcenter.
What the orthocenter is
An altitude runs from a vertex perpendicular to the opposite side (extended if necessary). All three altitudes of a triangle pass through one point — the orthocenter. This page writes two altitudes as linear equations and solves the 2×2 system exactly, so horizontal and vertical sides (which break slope-based methods) are handled cleanly.
Location is diagnostic: acute triangles keep the orthocenter inside, a right triangle parks it exactly on the right-angle vertex, and obtuse triangles push it outside entirely.
The Euler line
The orthocenter H, centroid G, and circumcenter O of any non-equilateral triangle are collinear — the Euler line — and G always sits one-third of the way from O to H, so H = 3G − 2O. This page uses that identity to hand you the other two centers free: for the default triangle, H(3, 3), G(3.667, 2.333), and O(4, 2) line up exactly, which is also a quick way to check any hand calculation.
How it’s calculated
The altitude from A satisfies (x₃ − x₂)x + (y₃ − y₂)y = (x₃ − x₂)x₁ + (y₃ − y₂)y₁ (perpendicular to BC through A); the altitude from B is built the same way from CA. Solving the two equations gives the orthocenter H. Centroid G = ((x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3); circumcenter from the Euler relation O = (3G − H)/2. Displayed to 3 decimals.
2D coordinates only; collinear points form no triangle, so no orthocenter is reported.
Where the orthocenter lands
| Triangle type | Orthocenter location |
|---|---|
| Acute | Inside the triangle |
| Right | Exactly at the right-angle vertex |
| Obtuse | Outside the triangle |
| Equilateral | Coincides with the centroid and circumcenter |
Classical Euclidean geometry — the three altitudes of any triangle are concurrent.
Common mistakes
- Confusing altitudes (perpendicular to the opposite side) with medians (to the midpoint) — medians meet at the centroid instead.
- Expecting the orthocenter inside every triangle; obtuse triangles put it outside.
- Hunting for a mystery point in a right triangle — it is exactly the right-angle vertex.
- Slope-based hand methods dividing by zero on horizontal or vertical sides; use perpendicular-vector equations instead.
Frequently asked questions
How do you find the orthocenter?
Write two altitude equations — each passes through a vertex and is perpendicular to the opposite side, e.g. (x₃ − x₂)x + (y₃ − y₂)y = (x₃ − x₂)x₁ + (y₃ − y₂)y₁ for the altitude from A — and solve them simultaneously. Their intersection is the orthocenter.
Where is the orthocenter of a right triangle?
Exactly at the right-angle vertex. The two legs are themselves altitudes, and they already meet there.
What is the Euler line?
The straight line through the orthocenter H, centroid G, and circumcenter O of any non-equilateral triangle. G divides the segment so that H = 3G − 2O — which is how this page derives the circumcenter it shows.
Can the orthocenter be outside the triangle?
Yes — whenever the triangle is obtuse. Two altitudes then meet the extensions of the opposite sides, and their crossing point lands outside the shape.