Find a triangle’s orthocenter — the point where all three altitudes intersect — from its three vertex coordinates.
How it works
Each altitude passes through one vertex, perpendicular to the opposite side. Solving two of the three altitude equations simultaneously finds their common intersection point. The triangle (0,0), (4,0), (1,3) has an orthocenter at (1, 1).
What this does not include
This finds the orthocenter only. It doesn’t report other triangle centers (like the centroid or circumcenter), which use different construction rules and generally land at different points.
How to use this calculator
- Enter the coordinates of all three vertices.
A worked example
A triangle with vertices (0,0), (4,0), (1,3): orthocenter = (1, 1) — where the triangle’s three altitudes intersect.
Vertices (0,0), (6,0), (2,4): orthocenter = (2, 2).
What the variables mean
| Variable | Meaning |
|---|---|
| (ax,ay), (bx,by), (cx,cy) | The triangle’s three vertex coordinates |
Edge cases worth knowing
The orthocenter can fall outside the triangle entirely for obtuse triangles — it only sits inside for acute triangles, a common surprise when first learning this construction.
Three collinear points don’t form a real triangle — there’s no meaningful altitude intersection for a degenerate “triangle” with zero area, so the calculator declines to show a result.
Can the orthocenter fall outside the triangle?
Yes — for an obtuse triangle, the orthocenter lies outside the triangle entirely; for a right triangle, it sits exactly at the right-angle vertex; only for an acute triangle does it fall inside.
How is the orthocenter different from the centroid?
The centroid is the intersection of the medians (lines from each vertex to the midpoint of the opposite side) and always lies inside the triangle — a completely different construction from the orthocenter’s altitude intersections.
What happens if the three points are collinear?
They don’t form a valid triangle at all, so there’s no orthocenter to find — this calculator returns no result in that case.
