Nine- Point CIrcle
Background Info and Mathematical Significance:
The nine-point circle is a circle that can be constructed for an triangle and is name that way because it passes 9 significant concyclic points. Which are the midpoints of each side of the triangle, the foot of each altitude and the midpoint of the line segment from each vertex to the orthocenter.
The Euler's Circle and the Feuerbach's circle are other names for the 9 Point Circle
Geogebra Activity
1. To find the center of a triangle's nine point center you first have to draw a triangle and then the midpoints of the three sides. Next you construct the feet of the altitudes of the triangle and through this they will intersect which is the orthocenter. After that you construct the midpoints. And through those point you have created the nine point excluding the orthocenter. Lastly to find the center you construct the circumscribe triangle for the triangle inside and there is your center.
2. The points that are located are the midpoints, the foot of each altitude and the midpoint of the line segment from each vertex to the orthocenter.
3. No, the center does not always lie inside the triangle.
4. Yes, it is possible for 2 or more of the nine points to overlap. This happens when it is an isosceles or right triangle