June 26, 2019

### 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.

the segment connecting the centroid to the apex is twice 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**.