A triangle is the simplest polygon there is — three points (vertices) and three lines (sides). Yet even this simple shape hides remarkably special points. Draw certain lines using its three sides or three angles, and remarkably, those lines always meet at exactly one point. Such a point is called a "center" of the triangle, and the five most notable ones together are known as the triangle's five centers.
The most intuitive one is the centroid. The line connecting each vertex to the midpoint of its opposite side is called a "median," and the three medians always meet at one point — that point is the centroid. Just as its name suggests, if you actually cut out a triangle-shaped board and balance it on a finger, it balances exactly at this point. The centroid also divides each median in a 2:1 ratio measured from the vertex, and that ratio stays exactly the same no matter the triangle's shape.
The circumcenter is the point where the perpendicular bisectors of all three sides meet. By definition, every point on a perpendicular bisector is equidistant from that side's two endpoints, so the circumcenter — where all three perpendicular bisectors meet — is equidistant from every side's endpoints, meaning all three vertices. Drawing a circle centered on the circumcenter with that distance as the radius produces a circle passing through all three vertices of the triangle (the circumscribed circle).
The incenter, in contrast, is where the three angle bisectors meet. A point on an angle bisector is equidistant from the two sides forming that angle, so the incenter — where all three angle bisectors meet — is equidistant from all three sides of the triangle. Drawing a circle centered on the incenter with that distance as the radius produces a circle that touches each of the three sides at exactly one point (the inscribed circle). Unlike the circumcenter, the incenter always lies inside the triangle, no matter its shape.
The orthocenter is the point where the three altitudes — lines dropped perpendicularly from each vertex to its opposite side — meet. This point's location shifts a lot depending on how "pointy" the triangle is: it sits inside an acute triangle, outside an obtuse triangle, and lands exactly on the vertex of the right angle for a right triangle.
Finally, the excenter may feel a little unfamiliar. It's the point where the internal bisector of one angle meets the external bisectors of the other two angles — and every triangle actually has three of these (one for each choice of which angle to bisect internally). An excenter is the center of a circle (an escribed circle) that touches one side of the triangle and the extensions of the other two sides.
Among these five points, three — the centroid (G), circumcenter (O), and orthocenter (H) — hold an even more striking property. No matter the triangle's shape, these three points always lie on a single straight line! This line is named the Euler line, after the mathematician who discovered it. What's more, the centroid divides the segment from the circumcenter to the orthocenter on that line in an exact 1:2 ratio. In an equilateral triangle, all three of these points (and even the incenter) collapse onto a single point, making it a special case where the Euler line itself seems to disappear.
On our activity page, you can freely drag a triangle's three vertices and see exactly how each of the five centers is constructed and how it moves as the triangle's shape changes — especially the moment when the circumcenter or orthocenter steps outside the triangle.