## Part 3

### I chose the discussion of the centers of a triangle as the representative of my best work. In the dicsussion, I defined the centroid, orthocenter, circumcenter, and incenter. I also showed the location of the different centers for various shapes of triangles. For each center, I also gave a connecting link to the Geometer's Sketchpad Script. I felt that a discussion of the Nine-Point Circle and the Euler Line were most appropriate for closing out the idea of centers of a triangle.

• In investigating the centers of triangles, I learned a lot about the interrelatedness of many aspects of mathematics. For example, the centroid of a triangle is the common intersection of the three medians. At the same time, it divides each median into two parts the ratio of whose lengths is 2 to 1. The six small triangles formed by the medians have equal triangles. Finally, the centroid plays a vital part in reconstructing the original triangle ABC when given the medians triangle. Many other mathematical ideas that I learned from this exploration are discussed in the paper "Centers of a Triangle".

• I was most satisfied with this paper because I learned a great deal about geometry and computers that I can use in the middle school classroom.

• I feel that this write-up was very thorough in that I connected the definitions, positions of various triangles, Geometer Sketchpad Scripts, and many other discoveries into one concise paper.

• New elements of mathematics were learned in every exploration. I felt that the Centers of a Triangle discussion was the one that I learned the most from because it was interesting to me. I tried to explore each problem to its fullest extent, even though I only chose to discuss a few of them. This assignment was a motivation to me in that it helped me to see that new areas of mathematics can be discovered by students with just a little bit of up-front guidance. I will be using the Geometer's Sketchpad as a tool for my students to discover the properties of mathematics.