# Orthocenter, Centroid, Circumcenter and Incenter of a Triangle

### Orthocenter

The **orthocenter** is the **point of intersection** of the **three heights** of a triangle.

A **height** is each of the **perpendicular lines** drawn from one **vertex to the opposite side** (or its extension).

### Centroid

The **centroid** is the point of intersection of the **three medians**.

A **median** is each of the **straight** **lines** that joins the **midpoint** of a side with the **opposite vertex**

The **centroid** divides each **median** into **two segments**, the segment joining the centroid to the vertex is twice the length of the length of the line segment joining the midpoint to the opposite side.

BG = 2GA

### Circumcenter

The **circumcenter** is the **point of intersection** of the **three perpendicular bisectors**.

A **perpendicular bisectors** of a triangle is each line drawn perpendicularly from its midpoint.

The **circumcenter** is the **center** of a triangle's circumcircle (circumscribed circle).

### Incenter

The **incenter** is the point of intersection of the three angle bisectors.

The angle bisectors of a triangle are each one of the lines that divide an angle into two equal angles.

The **incenter** is the center of the **circle inscribed** in the triangle.

### Line of Euler

The **orthocenter**, the **centroid** and the **circumcenter** of a non-equilateral **triangle are aligned**; that is to say, they belong to the same straight line, called **line of Euler**.