
Michelle E. Chung 
*
EMAT6680 Assignment 4: Centers of a 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:
 Let AB = segment between A and B.
 Let BC = segment between B and C.
 Let CA = segment between C and A.
 Let E = midpoint of AB.
 Let EC = line through midpoint E and point C.
 Let F = midpoint of BC.
 Let FA = line through midpoint D and point B.
 Let D = midpoint of CA.
 Let DB = line through midpoint D and point B.
 Let "G(Centroind)" = intersection of DB and EC.

Picture 


Location 


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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:
 Let AB = segment between A and B.
 Let BC = segment between B and C.
 Let CA = segment between C and A.
 Let l = line perpendicular to AB passing through C.
 Let j = line perpendicular to CA passing through B.
 Let k = line perpendicular to BC passing throuhg A.
 Let FA = line through midpoint D and point B.
 Let "H(Orthocenter)" = intersection of perpendicular line j and perpendicular line k.

Picture 


Location 


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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:
 Let CA = segment between C and A.
 Let AB = segment between A and B.
 Let BC = segment between B and C.
 Let D = midpoint of CA.
 Let l = line perpendicular to CA passing through D.
 Let E = midpoint of AB.
 Let j = line perpendicular to AB passing through E.
 Let F = midpoint of BC.
 Let k = line perpendicular to BC passing through F.
 Let "C(Circumcenter)" = intersection of perpendicular line j and perpendicular line k.

Picture 


Location 


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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:
 Let AB = segment between A and B.
 Let BC = segment between B and C.
 Let CA = segment between C and A.
 Let l = ray bisecting angle CAB.
 Let j = ray bisecting angle ABC.
 Let k = ray bisecting angle BCA.
 Let "I(Incenter)" = intersection of bisector j and bisector k.

Picture 


Location 


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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 
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 
Right Triangle 
Only Orthocenter, Centroid, and Circumcenter are on the same line. 
Centers of Right triangle 
Obtuse 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 
* The centers of a triangle is inside, on, or outside of the triangle, and orthocenter, centroid, and circumcenter are always on the same line, which is called 'Euler Line'. 
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. 




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