# UNIT 05: CENTERS OF TRIANGLES

## EXERCISE 06: 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**.

STEPS:

- Draw a triangle and call its vertices ABC.
- Draw the line bisector of side AB.
- Draw the line bisector of side BC.
- Draw the line bisector of side AC.
- The three line bisectors intersect at a point, which we call O, the circumcenter.
- If we make a circle centering the compass at the circumcenter and opening it to one of the vertices of the triangle, this circle will necessarily pass by the other two. This is the circumcircle of the triangle.

## 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**.

STEPS:

- Draw a triangle and call its vertices ABC.
- Draw the angle bisector of angle A.
- Draw the angle bisector of angle B.
- Draw the angle bisector of angle C.
- The three angle bisectors intersect at a point, which we call I, the incenter.
- Draw the perpendicular line, for example, to side AC passing through point I.
- If we make a circle centering the compass at the incenter and opening it to point T, this circle will necessarily be tangent to the other two sides. This is the incircle of the triangle.

To know the radious of the incircle and be able to draw it we need to know the least distance from the incenter to the sides of the triangle.

The least distance from a point to a line is the perpendicular.

If we draw the perpendicular line from the incenter to any of the sides of the triangle, we will get the radious of the incircle.

Where this line crosses side AC, we get T, the tangency point to this side.

## 3. CENTROID

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.

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

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

Barycenter is **always inside the triangle**.

STEPS:

- Draw a triangle and call its vertices ABC.
- Draw the perpendicular bisector of side AC, so you get the midpoint of this side MAC.
- Join MAC to the oposite vertex B, this is the median of vertex B.
- Draw the perpendicular bisector of side AB, so you get the midpoint of this side MAB.
- Join MAB to the oposite vertex C, this is the median of vertex C.
- Draw the perpendicular bisector of side BC, so you get the midpoint of this side MBC.
- Join MBC to the oposite vertex A, this is the median of vertex A.

The three medians intersect at a point, which we call G, the barycenter.

## 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**.

STEPS:

- Draw a triangle and call its vertices ABC.
- Draw the perpendicular line to side AC passing through point B. This is the B’s height.
- Draw the perpendicular line to side AB passing through point C. This is the C’s height.
- Draw the perpendicular line to side BC passing through point A. This is the A’s height.
- The three heights intersect at a point, which we call H, the orthocenter.

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