Two special centers hidden inside and outside a triangle
A triangle has two other special points besides the centroid: the circumcenter, which is equally distant from all three vertices, and the incenter, which is equally distant from all three sides. Move one vertex with the sliders to see how these two points move and why they have these properties.
Vertex C x-coordinate1
Vertex C y-coordinate4
Circumcenter: intersection of the perpendicular bisectors of the three sides
Inside the triangle
Why is the circumcenter equally distant from all three vertices? Every point on the perpendicular bisector of a segment is equally distant from the segment's two endpoints. The perpendicular bisectors of AB, BC, and CA meet at one point. That point is equally distant from A and B and equally distant from B and C, so it is ultimately equally distant from all three vertices A, B, and C. This point is the circumcenter, and the circle with this distance as its radius passes through all three vertices (the circumcircle).
In an acute triangle, the circumcenter is inside the triangle; in a right triangle, it is at the midpoint of the hypotenuse; and in an obtuse triangle, it lies outside the triangle.
Why is the incenter equally distant from all three sides? Every point on an angle bisector is equally distant from the two sides that form the angle. The three internal angle bisectors meet at one point, and that point is always equally distant from any pair of sides. This point is the incenter, and the circle with this distance as its radius is tangent to all three sides (the incircle).
Regardless of the triangle's shape, the incenter is always inside the triangle — unlike the circumcenter.