# Centroid, Incentre, Circumcentre & Orthocenter

### Centroid:

#### Definition:

**Coordinates:** The centroid of a triangle with vertices $({x}_{1},{y}_{1})$, $({x}_{2},{y}_{2})$, and $({x}_{3},{y}_{3})$ is given by: $(\frac{{x}_{1}+{x}_{2}+{x}_{3}}{3},\frac{{y}_{1}+{y}_{2}+{y}_{3}}{3})$
**Concept:** The centroid is the point of intersection of the medians of a triangle.
**Property:** Divides each median into segments with a ratio of 2:1.

### Incenter:

#### Definition:

**Coordinates:** The incenter of a triangle with vertices $({x}_{1},{y}_{1})$, $({x}_{2},{y}_{2})$ and $({x}_{3},{y}_{3})$ is given by: $(\frac{a{x}_{1}+b{x}_{2}+c{x}_{3}}{a+b+c},\frac{a{y}_{1}+b{y}_{2}+c{y}_{3}}{a+b+c})$ where $a,b,$ and $c$ are the side lengths opposite the respective vertices.
**Concept:** The incenter is the point of intersection of the angle bisectors of a triangle.
**Property:** Equidistant from the sides of the triangle.

### Circumcenter:

#### Definition:

**Coordinates:** The circumcenter of a triangle with vertices $({x}_{1},{y}_{1})$, $({x}_{2},{y}_{2})$, and $({x}_{3},{y}_{3})$ is given by: $(\frac{{D}_{1}}{2K},\frac{{D}_{2}}{2K})$ where ${D}_{1}$ and ${D}_{2}$ are certain determinants and $K$ is the area of the triangle.
**Concept:** The circumcenter is the point of intersection of the perpendicular bisectors of the sides of a triangle.
**Property:** Equidistant from the vertices of the triangle.

### Orthocenter:

#### Definition:

**Coordinates:** The orthocenter of a triangle with vertices $({x}_{1},{y}_{1})$, $({x}_{2},{y}_{2})$, and $({x}_{3},{y}_{3})$ is given by: $(\frac{{\sum}_{i=1}^{3}{x}_{i}({h}_{i}^{2}-{a}_{i}^{2})}{{\sum}_{i=1}^{3}({h}_{i}^{2}-{a}_{i}^{2})},\frac{{\sum}_{i=1}^{3}{y}_{i}({h}_{i}^{2}-{a}_{i}^{2})}{{\sum}_{i=1}^{3}({h}_{i}^{2}-{a}_{i}^{2})})$ where ${h}_{i}$ is the length of the altitude and ${a}_{i}$ is the length of the side opposite the $i$th vertex.
**Concept:** The orthocenter is the point of intersection of the altitudes of a triangle.
**Property:** May lie inside, outside, or on the triangle.

### Relationships:

**Equilateral Triangle:** For an equilateral triangle, all these points coincide at a single point.
**Different Types of Triangles:** The relative positions of these points may vary for different types of triangles (acute, obtuse, or right-angled).