I chose for my writeup #4 to be about problem 6:
Take any triangle. Construct a triangle connecting the three midpoints of the sides. This is called the MEDIAL triangle. It is similar to the original triangle and one-fourth of its area. Construct (the centroid) G, (the orthocenter) H, (the circumcenter) C, and (the incenter) I for this new triangle. Compare to G, H, C, and I in the original triangle.
To begin this exploration, I tried a few different types of examples to show that the medial triangle is in fact, similar to the original triangle and one fourth of its area for all types of triangles. Shown below, I explored this on the Geometers' Sketchpad for an acute, obtuse, and right triangle to confirm this works in all cases. If I did an activity exploring medial triangles in one of my future classrooms, I would likely show my students that these properties of the medial triangle hold for an acute triangle, and have them check to see if they work for any type of triangle.
As we can see in the above example, the corresponding angles for each pair of triangles are the same and the area of the medial triangles is one fourth the area of the original triangles in each case. Now that we have that information, we can explore the rest of the problem using any general triangle. I will choose to further my exploration using an acute triangle.
By construction, we can see that the centroid of the original triangle is in fact the same as the centroid! This makes a lot of sense when one considers the actual definition of the centroid and how it is constructed. The centroid is the point in which the three medians of the triangle meet one another. Since the medial triangle is constructed from the three midpoints of the triangle, it intuitively makes sense for the medians of the original triangles to pass through the midpoints of the medial triangle. For a rough sketch at a proof of this, one can rely on the properties we confirmed earlier about the medial triangles. Since it is similar to the original triangle, we can conclude that the midpoints of the medial triangle lie along the median lines of the original triangle.
Now we move on and consider the orthocenter of the two triangles. After construction (see below) one can see that these are not located in the same place as the centroid was; however each of the two points seem to lie in the same place on the each of the two triangles. Since these two triangles are similar, this is straightforward. In fact, if you measure the distances from the vertices to the orthocenter of the triangles, the ratios are the same when comparing the original triangle to the medial triangle, as shown in my picture.
When exploring the circumcenter of the triangles, one can also come up with some interesting proportions between the relationship of the original triangle and the medial triangle. The relationships formed in the above exploration of the proportions in the orthocenter would obviously hold in this case for the circumcenter, since by definition, the lengths of the segments formed from the vertices to the circumcenter are the same lengths. Interestingly enough though, if you construct a quadrilateral from the three vertices and the circumcenter for both the original and the medial triangle, both this area and the complement of this area have the same ratio to one another as the original triangle does to the medial triangle! For the original triangle ABC, the quadrilateral formed by ABC and the circumcenter is exactly four times the size of the quadrilateral formed by the vertices of the medial triangle and its circumcenter. This also holds for the complement of each of these regions. If you recall from the first information we were given about the medial triangles, the ratio of the area of the original triangle to the area of the medial triangle is also four to one. (See illustration below)
Lastly, we go into an exploration of the incenters of the two related triangles. The results end up being the same as the exploration of the circumcenters! Once the incenters of both triangles are found, one can create a quadrilateral formed by the vertices and incenter of the original triangle and also a quadrilateral formed by the vertices and incenter of the medial triangle. The ratio of the two resulting quadrilaterals is four to one, just like the ratio of the two triangles. Similarly, the ratio of the complements of the two regions is also four to one, as illustrated in the diagram below.
In conclusion, one can see that the medial triangle is special in that it shares many of the same properties of the original triangle. The common ratio of the two areas stays the same for many different regions using the same points of the two similar triangles. In a high school classroom, this could be a useful activity to combine several different concepts at the same time in a unique and interesting way. The students could investigate with this activity how to find the centroid, orthocenter, circumcenter, and the incenter of triangles and also see how certain properties of similar triangles are related to one another. Some partitions of the similar triangles have the same ratios as the triangles do, and for other properties of a triangle, there are even more relationships between the two.