6. The Centers of a Triangle

One can construct four centers of any triangle.

1. The centroid (G) is the intersection of the three medians of the triangle.

    

2. The orthocenter (H) is the intersection of the three lines containing the three altitudes of the triangle.

    

3. The circumcenter (C ) is the center of the circle circumscribed about the triangle. Therefore, C is the intersection of the perpendicular bisectors of the three sides of the triangle.

    

4. The incenter (I) is the center of the circle inscribed in the triangle. Therefore, I is the intersection of the angle bisectors of the three sides of the triangle.

One can use GSP to construct all four centers of a triangle ABC, and use the continuous variation of the software to observe relationships and make and test conjectures about relationships among these four centers of the triangle while manipulating the geometry of triangle ABC.

Hopefully, Here is a (will be) link to GSP sketch for interacting with the sketch

General Observations

Regardless of the shape of triangle ABCG, G, H, and C are collinear with G between H and C.

I is sometimes, but not always collinear with G,H, and C.

CG is one third of HC and HG is two thirds of HC.

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To refine these observations, adjust triangle ABC to fit a particular description and note the effect on G,H, C, and I. Let l be the line determined by G, H, and C.

Equilateral Triangle

G,H, C, and I coincide.

l contains a median/perpendicular bisector/altitude/angle bisector of triangle ABC.

Isosceles Triangle

l contains a median/perpendicular bisector/altitude/angle bisector from the vertex of the vertex angle to the base.

I lies on l.

Obtuse Triangle

H and C are outside the triangle

Acute Triangle

H and C are inside the triangle

Right Triangle

H is the vertex of the right angle.

C is the midpoint of the hypotenuse.

So, l contains the median from the vertex of the right angle to the hypotenuse.