# 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.

STEPS:

1. Draw a triangle and call its vertices ABC.
2. Draw the perpendicular bisector of side AB.
3. Draw the perpendicular bisector of side BC.
4. Draw the perpendicular bisector of side AC.
5. The three perpendicular bisectors intersect at a point, which we call O, the circumcenter.
6. If we make a circumference centering the compass at the circumcenter and opening it to one of the vertices of the triangle, this circumference will necessarily pass by the other two. This is the circumcircle of the triangle. With this measure draw an arc centering your compass in point C.

## 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.

STEPS:

1. Draw a triangle and call its vertices ABC.
2. Draw the angle bisector of angle A.
3. Draw the angle bisector of angle B.
4. Draw the angle bisector of angle C.
5. The three angle bisectors intersect at a point, which we call I, the incenter.
6. Draw the perpendicular line to side AC passing through point I.
7. Where this line crosses segment AC, we get T, the tangency point to this side.
8. If we make a circumference centering the compass at the incenter and opening it to point T, this circumference will necessarily be tangent to the other two sides. This is the incircle of 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.

STEPS:

1. Draw a triangle and call its vertices ABC.
2. Draw the perpendicular bisector of side AC, so you get the midpoint of this side MAC.
3. Join MAC to the oposite vertex B, this is the median of vertex B.
4. Draw the perpendicular bisector of side AB, so you get the midpoint of this side MAB.
5. Join MAB to the oposite vertex C, this is the median of vertex C.
6. Draw the perpendicular bisector of side BC, so you get the midpoint of this side MBC.
7. Join MBC to the oposite vertex A, this is the median of vertex A.
8. 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:

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

## SOLUTION OF DRAWING SHEET

Here you can see how to compose the complete drawing sheet, remember that you must underline the blue lines with your 0.4 technical pen, the red ones with 0.8 and the black ones with 0.2.