Investigating centers of a Triangle
and the Centroid of a Triangle
In this write-up, 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).
The interesting properties of the centroid
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
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
For an equilateral triangel, the circumcenter and the centroid are the same point(G).
* perpendicular bisectors (pink dished line)
* medians (green line)
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