I would use the GSP program with my geometry class when were discussing triangles. I would do one of two lessons depending on the level of the class and how comfortable I was with the intelligence and ability of the class to function in the computer lab and on the computers.
If the class was not as independent and needed a lot of probing I would have them explore triangles first for about ten minutes or so. Then we would begin to discuss the different types of triangles we could draw. From the triangles we could construct the perpindicular bisectors of the vertex and the opposite side and get the orthocenter (H). We can also work with angle bisectors and get the incenter (I)After that we would construct the midpoints of the triangles and discuss what we know about the midpoints and what we can find with the midpoints. From this we could draw the centroid (G) and the circumcenter (C). I would not just tell the students how to do this but I would have them explore with the triangles first and see if they could come up with some suggestions on their own. Then toward the end of the first class I would give them a couple of hints on how to find the orthocenter (H), centroid (G), circumcenter (C), and the incenter (I) and then have them look in their books and think about it for the next class. At the start of the next class I would ask if anyone had ideas and then from their proceed to tell them how they can find it. Then I would ask them to look at different triangles and how the four points are related. Then we would discuss this. After that I would proceed to discuss how a triangle inside the triangle we were working with would be related. This is probably as far as I would go with triangles in my geometry class, however we would look at the possibilities with all the triangles including scalene, isoscelels, equilateral, acute, obtuse, and right, and how each one varies. This introduction and exploration of triangles would probably last three class periods.
If the class was very independent and I thought that they could handle it I would give each member of the class a particular topic, such as the centroid of triangle and how that relates to an acute triangle within the triangle. I would then have the basic topics begin the discussion and work up to the more complicated topics. This would present the material but the students would be doing it so it would be more beneficial. I think that they would learn more coming from a peer, as long as there was good supervision. If the topics were greater than the students I would allow for the students to explore on their own and then introduce the basic vocabulary, such as orthocenter (H), centroid (G), circumcenter (C), and the incenter (I) and then give them topics from there.