# UNIT 05: 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.

Fisrt of all, we need to know what a point of concurrency is, because all the centers of triangles are

**points of concurrency**.

## Point of concurrency:

It is 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.

## Centers of triangles

### 1. CIRCUMCENTER

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

It is located at the intersection of the three side bisectors. As we already know the line bisector is the **locus of all the circle centers that pass through the endpoints of a segment**. As the three sides of a triangle are segments, if we draw the three side bisectors, we will get a point that will be the center of a circle that passes through the three vertices of the triangle, the Circumcircle.

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 circle that will fit inside the triangle. Each of the triangle’s three sides is a tangent to the circumference.

It is located at the intersection of the three angle bisectors. As we already know **the angle bisector is the locus of all the circle centers that are tangent to the sides of an angle**. As a triangle has three angles, if we draw the three angle bisectors, we will get a point that will be the center of a circle that is tangent to the three sides of the triangle, the Incircle.

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.

### To learn more…

*We have studied four centers of triangles, circumcenter, incenter, centroid and orthocenter. Orthocenter, circumcenter and centroid lay in the same line: the Euler line, named after Leonhard Euler, a Swiss mathematician. Euler line is a line determined from any triangle that is not equilateral. It passes through several important points determined from the triangle, including the orthocenter, the circumcenter, the centroid, the Exeter point and the center of the nine-point circle of the triangle.*

*Euler’s line:*

*Nine-point circle:*

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