U47 The Four Centers of a Triangle: Essential Formulas

Definition and Properties

The centroid $G$ is the intersection point of the three medians of a triangle. A median is a line segment from a vertex to the midpoint of the opposite side. The centroid divides each median in the ratio $2:1$, with the longer segment being from the vertex to the centroid.

Given vertices $A(x_1, y_1)$, $B(x_2, y_2)$, and $C(x_3, y_3)$, the coordinates of the centroid are as shown above.

Definition and Properties

The orthocenter $H$ is the intersection point of the three altitudes of a triangle. An altitude is a perpendicular line segment from a vertex to the line containing the opposite side (or to the side itself). In an acute triangle, the orthocenter lies inside the triangle; in an obtuse triangle, it lies outside.

There is no single simple coordinate formula like the centroid. It is typically found by solving the equations of any two altitudes. For vertex $A(x_1, y_1)$ with opposite side $BC$, the slope of the altitude from $A$ is the negative reciprocal of the slope of $BC$.

Definition and Properties

The circumcenter $O$ is the intersection point of the perpendicular bisectors of the three sides of a triangle. It is the center of the circumcircle (the circle that passes through all three vertices). The distance from $O$ to each vertex is the circumradius $R$.

Its coordinates can be found by solving the equations of any two perpendicular bisectors. The perpendicular bisector of a line segment has a slope that is the negative reciprocal of the segment's slope and passes through the segment's midpoint.

Where $a$, $b$, $c$ are the side lengths opposite vertices $A$, $B$, $C$ respectively, and $\Delta$ is the area of the triangle.

Definition and Properties

The incenter $I$ is the intersection point of the three angle bisectors of a triangle. It is the center of the incircle (the circle tangent to all three sides). The distance from $I$ to each side is the inradius $r$.

Given vertices $A(x_1, y_1)$, $B(x_2, y_2)$, $C(x_3, y_3)$ with opposite side lengths $a$, $b$, $c$ respectively. Also, the area $\Delta$ and inradius $r$ are related by $\Delta = r \cdot s$, where $s = \frac{a+b+c}{2}$ is the semi-perimeter.