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? Explore for many shapes of triangles.

Just for clarity:

G = Centroid; H = Orthocenter; C = Circumcenter; and I = Incenter.

Click Here for a link to a triangle with all four of these "centers"

And here is some information on what these terms mean, how to contruct them using GSP, as well as some observations:

__Centroid:__

The centroid of a triangle is often called the center of a triangle, however this is a vague definition. Instead, think of the centroid as the center of mass in the triangle; if the triangle were a 3D object and you could hold it, then you could balance it on your finger at the centroid. Therefore, the centroid must always be inside the triangle.

Start by drawing a triangle, then highlight a line and construct the midpoint. Do this for all three lines. Next, highlight a line and the opposite vertex and construct a segment. Again, do this for the remaining lines and vertices. The amazing result of this is that all three lines intersect at the same point! This point is the centroid.

__Circumcenter:__

The circumcenter is the point associated with a triangle that is equidistant from all three vertices. Because of the various sizes and shapes of triangles, this center doesn't always have to be located inside the triangle. Indeed, once an obtuse angle is present in a triangle, the circumcenter appears outside of the triangle on the opposite side of the obtuse angle. Furthermore, the circumcenter lies on the hypotenuse of a right triangle.

To begin the process of constructing the circumcenter, the midpoints of all three segments must be constructed. Then you construct the perpendicular bisector to each side through the midpoint. Once again, all three of these lines will intersect at the same point--the circumcenter.

__Orthocenter:__

The orthocenter is the intersection of all three altitudes in a triangle. To construct an altitude, just highlight a vertex and the line opposite it and construct a perpendicular line. Do this for all three sides, and the intersection is the orthocenter. Like the circumcenter, the orthocenter can be inside or outside the triangle. However, it can only "leave" the triangle by way of a vertex; a very interesting fact.

__Incenter:__

The incenter of a circle is the intersection of all three angle bisectors. Because angle bisectors pass through the interior of a triangle, we know that the incenter must always lie inside the triangle.

To construct the incenter, select three vertices which will represent an angle. Then construct an angle bisector for that angle. Continue this process with the other two angles. The intersection of these lines is the incenter.

__General Observations:__

One of the most amazing facts about G, H, and C is that they all lie on the same line! Not only that, but there is always the same ratio of distance between them. To be more precise, the distance from the centroid to the orthocenter is twice that of the distance from the centroid to the circumcenter. No matter what type of triangle being used--obtuse, equilateral, etc.--these phenomenons are always present.

Also, all four centers are one in an equilateral triangle. They also all four line up on the same line in an isosceles triangle. Furthermore, the area of triangle HIG is twice the area of triangle CGI, no matter what type of triangle the original one is.