Let's construct a triangle and its medial triangle. The medial triangle of a triangle is created by constructing segments between the midpoints of each side of the original triangle. In the construction below, we have labeled the centroid, orthocenter, circumcenter, and incenter of each triangle as G, H, C, and I respectively. Each type of center is followed by the number 1 if it relates to the larger triangle, and is followed by a 2 if it relates to the medial triangle.
Some of the most notable things we can observe about these centers are:
- The centroid of both the original triangle and the medial triangle exist at the same point.
- The circumcenter of the original triangle and the orthocenter of the medial triangle exist at the same point.
- The distinct incenters of each triangle are collinear with the point at which the centroids of both triangles exist.
- The orthocenter of the original triangle, the circumcenter of the medial triangle, and the point at which the circumcenter of the original triangle and the orthocenter of the medial triangle exist are collinear.
- The circumcenter of the medial triangle divides the segment from the orthocenter of the original triangle to the point at which the circumcenter of the original triangle and orthocenter of the medial triangle exist into two segments of equal length.
- The segment between the incenters of the triangles is divided into two segments by the point where the centroids exist. The segment having an endpoint which is the incenter of the medial triangle is half the length of the segment having an endpoint which is the incenter of the original triangle.
Each of these bullet points presents an area which would make for good student discussion in a classroom setting.
Let's explore the second bullet point in more detail:
"The circumcenter of the original triangle and the orthocenter of the medial triangle exist at the same point."
First, let's consider the circumcenter of the original triangle. This is the point which is equidistant from each of the vertecies of the original triangle. More practically, this point is the center of a circle which would pass through each of the vertecies of the triangle. This point was created by determining the intersection point of the perpendicular bisectors of the sides of the original triangle.
Second, we'll consider the orthocenter of the medial triangle. This point is the point of intersection of the altitudes of the medial triangle.
The altitude of any triangle is a line through a vertex of the triangle which is perpendicular to the triangle side which is opposite that particular vertex.
So, if we look carefully, we can already notice that both the altitudes of the medial triangle and the perpendicular bisectors of the sides of the original triangle go through vertecies of the medial triangle. The altitudes of the medial triangle do this by definition. The perpendular bisectors of the original triangle go through the the midpoints of each side of the original triangle. Incidentally enough, these are the points at which the vertecies of the medial triangle (by definition) exist as well.
Through further investigation, we can also determine that both the altitudes of the medial triangle and the perpendicular bisectors of the sides of the original triangle are perpendicular to the sides of the original triangle whose midpoints they traverse. Perpendicular bisectors of the original triangle's sides are this way by definition. The altitudes of the medial triangle are each perpendicular to the side opposite the vertex of the medial triangle through which they each pass. Since each side of the medial triangle bisects two sides of the original triangle, each side of the medial triangle is parallel to the side of the original triangle on which its opposite vertex rests. Then, since the altitudes of the medial triangle are perpendicular to the sides opposite the vertecies through which they pass, and each side of the medial triangle is parallel to the side of the original triangle on which its oppostie vertex rests, we can say that the altitudes of the medial triangle are perpendicular to the side of the original triangle on which the vertex through which the altitude passes is lying.
We have now shown that both the altitudes of the medial triangle and the perpendicular bisectors of the sides of the original triangle are lines which bisect the sides of the original triangle, and that both the altitudes of the medial triangle and the perpendicular bisectors of the sides of the original triangle are perpendicular to the sides of the original triangle through which they pass. Then, both the altitudes of the medial triangle and the perpendicular bisectors of the sides of the original triangle are sets of perpendicular biesctors of sides of the original triangle, and are therefore the same two sets of three lines.
Since these lines are essentially two sets of the same lines, each point of intersection for both of these sets must be the same point.
Therefore, the circumcenter of the original triangle and the orthocenter of the medial triangle exist at the same point.
