Free Orthocenter Calculator

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The orthocenter is one of a triangle's four classic "triangle centers," found where its three altitudes — the perpendicular segments dropped from each vertex to the line containing the opposite side — all meet at a single point. It's a staple of high school and college geometry courses, and it appears alongside the centroid, circumcenter, and incenter in the study of triangle geometry; notably, the orthocenter, centroid, and circumcenter always lie on a single straight line called the Euler line.

How it works

Enter the triangle's three vertices. The calculator finds two altitude lines: each altitude passes through one vertex and is perpendicular to the side opposite that vertex, which means the direction vector of the opposite side becomes the normal vector of that altitude line. Setting up the equations for two of these three altitudes and solving them simultaneously gives the point where they cross — which, by the defining property of the orthocenter, is also exactly where the third altitude passes through.

Enter vertex A — x1.
Enter vertex A — y1.
Enter vertex B — x2.
Enter vertex B — y2.
Enter vertex C — x3.
Enter vertex C — y3.
Click Calculate to see your results.

Examples

(0,0), (4,0), (0,3)

This right triangle has its right angle at (0,0), so the two legs are themselves already altitudes — meeting exactly at that vertex. The orthocenter is (0, 0).

(0,0), (6,0), (3,6)

This isosceles triangle is symmetric about x=3, so its orthocenter lies on that line of symmetry. Working out the altitude from (3,6) straight down to the base, and the altitude from (0,0) perpendicular to the opposite side, gives an orthocenter of (3, 1.5).

Common mistakes to avoid

Entering three collinear points, which don't form a real triangle and have no orthocenter (the "altitudes" would all be parallel or undefined).
Confusing the orthocenter with the centroid or circumcenter — they are three different points defined by three different constructions.
Assuming the orthocenter is always inside the triangle — that's only guaranteed for acute triangles.

Frequently asked questions

It falls outside the triangle entirely — only acute triangles have their orthocenter strictly inside; right triangles have it exactly at the right-angle vertex.

It's the single straight line that always passes through a triangle's orthocenter, centroid, and circumcenter (unless the triangle is equilateral, in which case all three points coincide at the same spot).

The centroid is the intersection of the three medians (vertex to opposite side's midpoint) and always lies inside the triangle; the orthocenter is the intersection of the three altitudes (vertex to opposite side, perpendicular) and can lie inside, on, or outside the triangle depending on its shape.

All four classic triangle centers — orthocenter, centroid, circumcenter, and incenter — collapse onto the exact same point, due to the triangle's full symmetry.

Yes — this is a proven geometric fact for every triangle, no matter its shape, which is exactly what makes the orthocenter a well-defined single point rather than a vague approximation.