neunPunkteDiagramm
NeunPunkteDiagramm, also known as the Nine Points Circle, is a geometric construction used in Euclidean geometry to find the center of a given circle. The method is attributed to the German mathematician Karl Wilhelm Feuerbach, who published it in 1822. The Nine Points Circle is significant because it passes through nine specific points related to a given triangle, making it a useful tool in triangle geometry.
The nine points are as follows:
1. The midpoints of the three sides of the triangle.
2. The feet of the three altitudes (the points where the altitudes intersect the opposite sides).
3. The midpoints of the line segments joining each vertex of the triangle to the orthocenter (the
To construct the Nine Points Circle, one can use the following steps:
1. Construct the orthocenter of the triangle.
2. Draw the nine specific points mentioned above.
3. Construct the circumcircle of the orthic triangle (the triangle formed by the feet of the altitudes).
4. The Nine Points Circle is the nine-point circle of the orthic triangle.
The Nine Points Circle has several interesting properties, including:
- It has the same radius as the circumcircle of the medial triangle (the triangle formed by
- It is tangent to the incircle and the excircles of the original triangle.
- It has a power of -27 with respect to the original triangle's circumcircle.
The Nine Points Circle is a fundamental concept in Euclidean geometry and has applications in various