Investigating centers of a Triangle
The Circumcenter
and the Centroid of a Triangle
Introduction
In this writeup, I examine some centers of a triangle; Thecircumcenter and the centroidof a triangle.
First, I construct the circumcenter and the centroid of a triangle using GSP. Then, I will explore each center's location for shapes of triangles and properties. Finally, I investigate the relation between the circumcenter and the centroid of a triangle.
The circumcenter of a triangele is the intersection point of three perpednicular bisectors of each side in a triangle.
Since triangles APC and BPC are congruent(SAS), triangles BNC and DNC (SAS) and triangles AMC and DMC, AC=BC=DC.
So, any circle with C as center and passing through one of the vertices will go through all of them.
The circle is called circumscribed circle(circumcircle).
Inside  On side  Outside  
Right triangle 


Acute triangle 


Obtuse triangle 


Scalene triangle 



Isosceles triangle 



Equilateral triangle 

Inside  Onside  Outside  
Right triangle  x  
Acute triangle  x  
Obtuse triangle  x  
Scalene triangle  x  
Isosceles triangle  x  
Equilateral triangle  x 
The interesting properties of the centroid
1) Distance
Centroid in a triangle divides each median into two parts,
the ratio whose lengths is 2:1.
AF : FE = BF : FG = CF : FD = 2:1
2) Area
The six areas of the interior triangles formed by the
three medians in the triangle are all the same.
area CEF= area GFC =area AFG= area = ADF= area FEB= area
DBF
For an equilateral triangel, the circumcenter and the centroid are the same point(G).
* perpendicular bisectors (pink dished line)
* medians (green line)