# Centroid, Incentre, Circumcentre & Orthocenter

### Centroid:

#### Definition:

• Coordinates: The centroid of a triangle with vertices $\left({x}_{1},{y}_{1}\right)$, $\left({x}_{2},{y}_{2}\right)$, and $\left({x}_{3},{y}_{3}\right)$ is given by: $\left(\frac{{x}_{1}+{x}_{2}+{x}_{3}}{3},\frac{{y}_{1}+{y}_{2}+{y}_{3}}{3}\right)$
• 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 $\left({x}_{1},{y}_{1}\right)$, $\left({x}_{2},{y}_{2}\right)$ and $\left({x}_{3},{y}_{3}\right)$ is given by: $\left(\frac{a{x}_{1}+b{x}_{2}+c{x}_{3}}{a+b+c},\frac{a{y}_{1}+b{y}_{2}+c{y}_{3}}{a+b+c}\right)$ 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 $\left({x}_{1},{y}_{1}\right)$, $\left({x}_{2},{y}_{2}\right)$, and $\left({x}_{3},{y}_{3}\right)$ is given by: $\left(\frac{{D}_{1}}{2K},\frac{{D}_{2}}{2K}\right)$ 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 $\left({x}_{1},{y}_{1}\right)$, $\left({x}_{2},{y}_{2}\right)$, and $\left({x}_{3},{y}_{3}\right)$ is given by: $\left(\frac{{\sum }_{i=1}^{3}{x}_{i}\left({h}_{i}^{2}-{a}_{i}^{2}\right)}{{\sum }_{i=1}^{3}\left({h}_{i}^{2}-{a}_{i}^{2}\right)},\frac{{\sum }_{i=1}^{3}{y}_{i}\left({h}_{i}^{2}-{a}_{i}^{2}\right)}{{\sum }_{i=1}^{3}\left({h}_{i}^{2}-{a}_{i}^{2}\right)}\right)$ 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).