Lets begin this exploration of triangles by constructing a traiangle and its medial. A medial is a triangle constructed from using the midpoints of another triangle. This medial triangle is similar to teh original and one fourth its area. Below is a triangle with its medial:
Here the green triangle, we will call it traingle 1 for labeling purposes, is the medial of the blue triangle, the original triangle 0. Next lets discuss a few special points of a triangle. First there is the centroid (G) that is the common intersection of the three segments connecting the midpoint to the oppostie vertex, also called the median. Next is the orthocenter (H) which is the point of common intersection of the three perpendicular segments connecting the vertex to the opposite side, also called altitudes. The circumcenter (C) is the point in the plane that is equidistant from the three vertices of the triangle. Finally there is the incenter (I), a point in the interior of the triangle that is equidistant from the three sides. Below is a sketch of the original triangle with its four special points labeled:
The next sketch has the four points labeled for the medial triangle:
Now when we overlap the two sketches, we arrive at an a interesting sketch:
As we can see by the picture, there are ony 6 points showing inside the triangles when we plotted 8. Therefore these two triangles share 2 different points for thier special points. As shown above, both triangles have the same centroid, points G0 and G1, because since the medial is constructed from segments joining the midpoints of the bigger triangle. Henceforth, when a vertex and midpoint of the medial triangle is connected, if that liine were to be extended it would intersect the vertex of the bigger triangle opposite the vertex being used. Lets explore exactly what is happening with a sketch.
Since the centroid is constructed by the intersection of a triangle's vertex and its opposite midpoint, given that the medial triangle's vertexes are the bigger triangles midpoints, then the medial triangle's midpoints are collinear with vertexes and midpoints of the bigger traiangle. Therefore the line connecting a vertex and its opposite midpoint is the same line as a line connect that same midpoint, hence vertex of the medial, and the medials opposite midpoint.
Now lets explore the next point in the sketch that represents two different points. This point is the circumcenter of the bigger triangle and the orthocenter of the medial triangle. This means that, by the defintions of circumcenter and orthocenter, that the point in the plane that is equidistant from the 3 vertices of the bigger triangle is equal in some sense to the intersection of the altitudes of the medial. In other words, the perpendiular bisector of the bigger triangle are the same lines in the same plane as the perpendicular lines connecting the vertices and opposite sides of the medial. Lets explore that sketch.
As we can see, the orangle dotted lines iintersect the vertices of the medial and are perpendicular to its opposite side. Also, the lines are perpendicular bisectors at teh midpoints of each side of the bigger triangle. Therefore we can come to the conclusion that for any triangle and its medial, if the circumcenter of the bigger triangle and the orthocenter of the medial are constructed, they will always be the same point, no matter what the shape of the triangles are.