Explorations with Triangle Centers

Let's examine the centers of a triangle. We will start with an arbitrary triangle TriABC with it's sides extended (we'll see why later) in GSP.

Triangle ABC

Using GSP, we can simply select the sides of our triangle, and then select Construct Midpionts to obtain m1, m2, and m3 as shown.

Triangle ABC with midpoints added

Now we can select a vertex, say A, and the midpoint of the opposite side, m2, and construct the segment between them. Do likewise for B and m3, and C and m1. The three segments are concurrent and their intersection is called the centroid of TriABC. We label this G by convention.
(Note: the GSP construction is not a proof.)

Triangle ABC with midpoints, segments, and G

Now we can select our midpoints and our segments and "hide" them, in order to clean up our triangle before we move on. TriABC should now look like:

Triangle ABC with G

The next step is to select a vertex, say A, and the line opposite A, which is the extended line through BC, and constructing a perpendicular line from A through BC. Note that the intersection lies outside of TriABC. Repeat the process for the other two vertices. The three lines are concurrent and their intersection is called the orthocenter of TriABC. We label this H by convention.
(Note: the GSP construction is not a proof.)

Triangle ABC with G, perpendiculars, and H

Now we can select our perpendiculars and "hide" them, in order to clean up our triangle before we move on. TriABC should now look like:

Triangle ABC with G and H

For our next step, first reconstruct the midpoints of our segments (just like before). Then select a midpoint, say m1, and the segment the m1 lies on, in this case AB. Once selected construct a perpendicular line using GSP. This is called the perpendicular bisector of AB. Repeat the process for the other two segments. The three perpendicular bisectors are concurrent and their intersection is called the circumcenter of TriABC. Here we label it C also.
(Note: the GSP construction is not a proof... just want this point to stick!)

Triangle ABC with G, H, perpendicular bisectors, and C

Why is our newest point called the circumcenter? Because if we select the circumcenter to be the center of a circle whose radius is the length from the circumcenter to one of the vertices of TriABC, we obtain the circle that circumscribes TriABC, called the circumcircle. The picture below illustrates (it was cleaned first, can you duplicate it?).

Triangle ABC with G, H, C, and circumcircle

The keen eye may notice that G, H, and C appear to be colinear. In fact, they are. The following picture illustrates what is called the Euler line segment for TriABC. Can you construct the line segment?
(Note: the GSP construction is not a proof)

Triangle ABC with Euler line segment

Our final construction, is to construct the angle bisectors of each angle of TriABC. The angle bisectors will be concurrent and their intersection is called the incenter of TriABC, denoted I.
(Note: the GSP construction is still not a proof.)

Triangle ABC with Euler line segment and I

The following pictures illustrates why I is called the incenter. It is the center of the circle that is inside TriABC and is tangent to each side. Can you construct the incircle?

Triangle ABC with Euler line segment, I, and incircle

The following illustrates the relationships between the triangle centers for various triangles.

Various Triangles with G, H, C, and I

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Write-up by Robert Childres.