# Michelle E. Chung

* EMAT6680 Assignment 4: Centers of a Triangle

 CENTROID(G) ORTHOCENTER(H) CIRCUMCENTER(C) INCENTER(I) CENTERS OF TRIANGLE

1. The CENTROID(G) of a triangle is the common intersection of the three medians.
A median of a triangle is the segment from a vertex to the midpoint of the opposite side.
Let's construct the Centroid using Geometer's Sketchpad (GSP) and explore its location for various shapes.

Steps for constructing

Given:

Point A, Point B, Point C

Stpes:

1. Let AB = segment between A and B.
2. Let BC = segment between B and C.
3. Let CA = segment between C and A.
4. Let E = midpoint of AB.
5. Let EC = line through midpoint E and point C.
6. Let F = midpoint of BC.
7. Let FA = line through midpoint D and point B.
8. Let D = midpoint of CA.
9. Let DB = line through midpoint D and point B.
10. Let "G(Centroind)" = intersection of DB and EC.

Picture
Location
Acute Triangle Right Triangle The centroid of a acute triangle is inside of the triangle. Centroid of Acute triangle The centroid of a right triangle is inside of the triangle. Centroid of Right triangle The centroid of a obtuse triangle is inside of the triangle. Centroid of Obtuse triangle Centroid of triangle Movie

Back to the Top

2. The ORTHOCENTER(H) of a triangle is the common intersection of the three lines containig the altitudes.
An altitude is a perpendicular segment from a vertex to the line of the opposite side.
(Note: the foot of the perpendicular may be on the extention of the side of the triangle.)
It should be on clear that H does not have to be on the segments that are the altitudes.
Rather, H lies on the lines extended along the altitudes.

Steps for constructing

Given:

Point A, Point B, Point C

Stpes:

1. Let AB = segment between A and B.
2. Let BC = segment between B and C.
3. Let CA = segment between C and A.
4. Let l = line perpendicular to AB passing through C.
5. Let j = line perpendicular to CA passing through B.
6. Let k = line perpendicular to BC passing throuhg A.
7. Let FA = line through midpoint D and point B.
8. Let "H(Orthocenter)" = intersection of perpendicular line j and perpendicular line k.

Picture
Location
Acute Triangle Right Triangle The orthocenter of a acute triangle is inside of the triangle. Orthocenter of Acute triangle The orthocenter of a right triangle lies on the intersection of two sides (not hypotenuse). Orthocenter of Right triangle The orthocenter of a obtuse triangle is outside of the triangle. Orthocenter of Obtuse triangle Orthocenter of triangle Movie

Back to the Top

3. The CIRCUMCENTER(C) of a triangle is the point in the plane equidistant from the three vertices of the triangle.
Since a point equidistance from two points lies on the perpendicular bisector of the segment determined by the two points,
C is on the perpendicular bisector of each side if the triangle.
(Note: C may be outside of the triangle.)
Also, it is the center of the CIRCUMCIRCLE(the circumscribed circle) of the triangle.

Steps for constructing

Given:

Point A, Point B, Point C

Stpes:

1. Let CA = segment between C and A.
2. Let AB = segment between A and B.
3. Let BC = segment between B and C.
4. Let D = midpoint of CA.
5. Let l = line perpendicular to CA passing through D.
6. Let E = midpoint of AB.
7. Let j = line perpendicular to AB passing through E.
8. Let F = midpoint of BC.
9. Let k = line perpendicular to BC passing through F.
10. Let "C(Circumcenter)" = intersection of perpendicular line j and perpendicular line k.

Picture
Location
Acute Triangle Right Triangle The circumcenter of a acute triangle is inside, on, or outside of the triangle. Orthocenter of Acute triangle The circumcenter of a right triangle lies exactly at the midpoint of the hypotenuse (longest side). Orthocenter of RIght triangle The circumcenter of a obtuse triangle is always outside of the triangle. Orthocenter of Obtuse triangle Orthocenter of triangle Movie

Back to the Top

4. The INCENTER(I) of a triangle is the point on the interior of the triangle that is equidistant from the three sides.
Since a point interior to an angle that is equidistant from two sides of the angle lies on the angle bisector,
I must be on the angle bisector of each angle of the triangle.
Also, it is the center of the INCIRCLE(the inscribed circle) of the triangle.

Steps for constructing

Given:

Point A, Point B, Point C

Stpes:

1. Let AB = segment between A and B.
2. Let BC = segment between B and C.
3. Let CA = segment between C and A.
4. Let l = ray bisecting angle C-A-B.
5. Let j = ray bisecting angle A-B-C.
6. Let k = ray bisecting angle B-C-A.
7. Let "I(Incenter)" = intersection of bisector j and bisector k.

Picture
Location
Acute Triangle Right Triangle The incenter of a acute triangle is inside of the triangle. Incenter of Acute triangle The incenter of a right triangle is inside of the triangle. Incenter of Right triangle The incenter of a obtuse triangle is inside of the triangle. Incenter of Obtuse triangle Incenter of triangle Movie

Back to the Top

5. Use GSP to construct G, H, C, and I for the same triangle.
What relationships can you find among G, H, C, and I or subsets of them?

Picture

Location
Acute Triangle Right Triangle The four centers of a acute triangle is inside, on, or outside of the triangle, and all of them could be on the same line. Centers of Acute triangle Only Orthocenter, Centroid, and Circumcenter are on the same line. Centers of Right triangle The four centers of a acute triangle is inside or outside of the triangle, and all of them could be on the same line. Centers of Obtuse triangle Centers of triangle Movie

* On Euler line, there are Center of nine point circle, Circumcenter, Centroid, and Orthocenter. Also, the nine point circle exsits only on acute triangles.

Go Back to Michelle's Main page
Go Back to EMAT 6680 Homepage