Let us examine some important points related to a triangle. Use GeoGebra or any geometry software to follow the construction below.
1.) Construct triangle TUV.
2.) Construct the midpoint of each side.
3.) Construct the three altitudes. What do you observe? The intersection of the altitudes is called the orthocenter.
4.) Construct the three midpoints between the orthocenter and the three vertices. What do you observe about the nine points?
After finishing the steps above, your figure should at look like Figure 1. For the sake of discussion, we color the points. The red points are the midpoints of each side, the green points are the ‘foot’ of each altitude, and the cyan points are the midpoints of the orthocenter and and the vertices.
From the figure above, we observe two things: first, the altitudes seem to meet at a point; and second, it seems that the 9 points form a circle (can you verify this by construction?). » Read more