A triangle has four classical centres, and they usually do not coincide. The centroid is the vertex average; the circumcentre is equidistant from all three vertices; the orthocentre is where the altitudes meet; the incentre is where the angle bisectors do. Only in an equilateral triangle are all four the same point.
Four centres, four constructions
Each centre answers a different question, which is why there are four of them.
The centroid is where the medians meet, and it is the balance point — cut the triangle from card and it balances on a pin there. It is simply the average of the three vertices, which makes it the only one needing no construction at all.
The circumcentre is equidistant from all three vertices, so it is the centre of the circle through them. It is where the perpendicular bisectors meet.
The orthocentre is where the three altitudes meet. It has no distance property — it is defined purely by perpendicularity, which is why it can end up a long way from the triangle.
The incentre is equidistant from all three sides, so it is the centre of the circle that fits inside touching each one. It is where the angle bisectors meet.
In an equilateral triangle all four collapse to a single point. In every other triangle at least some of them separate.
The Euler line
Three of the four are always collinear, and that is not obvious at all.
The centroid, circumcentre and orthocentre lie on one straight line — Euler’s line — whatever the triangle. Move a vertex anywhere and the three of them move together, staying in line.
More than that: the centroid always sits exactly twice as far from the orthocentre as from the circumcentre. A fixed 2:1 ratio, for every triangle there is.
This page computes that ratio and displays it, because it verifies all three centres at once. If any one of them were computed wrongly the ratio would not come out at 2, so the check is meaningful rather than decorative.
The incentre is the exception. It joins the line only when the triangle is isosceles — and in that case the line is the axis of symmetry, so everything lands on it. Euler published this in 1765, and it remains one of the more surprising facts about a shape everyone thinks is fully understood.
When a centre leaves the triangle
Two of the four can end up outside the shape entirely.
The circumcentre is inside for an acute triangle, exactly on the hypotenuse midpoint for a right one, and outside for an obtuse one.
The orthocentre does the same, and more dramatically: it lands on the right-angled vertex in a right triangle, and can be far outside an obtuse one.
The centroid and incentre are always inside, whatever the shape. The centroid is an average of points inside, and the incentre is the centre of a circle that fits within.
The right-triangle cases are worth memorising as checks. The circumcentre on the hypotenuse midpoint is Thales’ theorem restated: any angle inscribed in a semicircle is a right angle.
The nine-point circle
There is a circle through nine notable points of any triangle, and its centre is the midpoint of the orthocentre and the circumcentre.
The nine: the three side midpoints, the three feet of the altitudes, and the three midpoints between each vertex and the orthocentre.
Its radius is exactly half the circumradius, and its centre also lies on the Euler line — making four notable points on that one line rather than three.
Feuerbach proved something further still in 1822: this circle is tangent to the incircle and to all three excircles. It is one of the results that make plane geometry feel less finished than it looks.
Where it gets used
Balance and structure. The centroid is the centre of mass of a uniform triangular plate, which is where a support has to go.
Circles through three points. The circumcentre is how you find the unique circle through three given points — used in surveying, in fitting an arc through measurements, and in Delaunay triangulation.
Meshing and graphics. Triangle centres are used to subdivide meshes and to test triangle quality, since a triangle whose circumcentre is far outside it is a poorly shaped one.
Competition geometry. Olympiad problems lean heavily on these centres and the relations between them, the Euler line most of all.
Sources and methodology
The four centres and the Euler line are classical geometry; these are the references.
Method. The four centres are computed independently and then checked against each other rather than merely reported. The centroid, circumcentre and orthocentre must be collinear with the centroid dividing that segment in a 2:1 ratio, and the page computes the ratio and shows it — a single wrong centre breaks it visibly. The suite asserts that ratio on a thousand generated triangles, and separately checks that the circumcentre is genuinely equidistant from all three vertices. That engine is verified on every change against 114 hand-written assertions, including that the centroid divides the Euler line in a 2:1 ratio on every one of a thousand generated triangles, and that the circumcentre is equidistant from all three vertices. The count and the per-case breakdown are published on the formula verification page.
Read the guide
To solve a triangle from side lengths and angles rather than coordinates, the Triangle Calculator covers every case including the ambiguous one.