# UNIT 06: TRIANGLES

## THE CENTERS OF TRIANGLES

## Journey to the Center of a Triangle

Here you can see a film, ‘Journey to the Center of a Triangle’ (1976), created by Bruce & Katharine Cornwell to help you understanding basic ideas about the centers of triangles.

The film created on the Tektronics 4051 Graphics Terminal. Presents a series of animated constructions that determine the center of a variety of triangles, including such centers as **circumcenter**, **incenter**, **centroid** and **orthocenter**. This movie is part of the collection: Academic Film Archive of North America.

**Points of concurrency:**The point where three or more lines intersect.

In the figure above the three lines all intersect at the same point P – called the point of concurrency.

## 1. CIRCUMCENTER

The **circumcenter** is the center of the **triangle’s circumcircle**, one that passes through the three vertices of the triangle.

Located at the intersection of the three side bisectors.

It is called graphically with the letter **O**.

- In an
**acute triangle**, the circumcircle’s center is inside the triangle. - In an
**obtuse triangle**, the circumcircle’s center is outside the triangle. - In a
**right triangle**, the circumcircle’s center is the midpoint of the hypotenuse.

## 2. INCENTER

The **incenter** is the center of the **triangle’s incircle**, also known as inscribed circle, it is the largest circumference that will fit inside the triangle. Each of the triangle’s three sides is a tangent to the circumference.

Located at the intersection of the three angle bisectors.

It is called graphically with the letter **I**.

Incenter is always **inside the triangle**.

## 3. BARYCENTER

The **barycenter** or **centroid** is located on the intersection of the three **medians** of the triangle, and is equivalent to the **center of gravity** of the triangle.

The **medians** of the triangle are each segments which join each vertex of the triangle with the midpoint of the opposite side.

It is called graphically with the letter **G**.

Barycenter is always **inside the triangle**.

## 4. ORTHOCENTER

The **Orthocenter** is the point located at the intersection of the triangle **heights**.

It is called the **height** of a triangle the segment which join a vertex of a triangle to the opposite side -or its extension- forming a right angle (90 °). All triangles have three heights.

It is called graphically with the letter **H**.

- In an
**acute triangle**, the orthocenter is inside the triangle. - In an
**obtuse triangle**, the orthocenter is outside the triangle. - In a
**right triangle**, the orthocenter is the vertex of the triangle’s right angle.

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