Nine-point circle

The Feuerbach circle or nine-point circle is a special circle in a triangle, which is named after Karl Wilhelm Feuerbach. Lying On him nine excellent points:

  • The midpoints of the sides;
  • The bases of the heights;
  • The centers of the upper level sections (which are the midpoints of the segments between each corner of the triangle and the orthocenter of the triangle ).

In the picture on the right are D, E and F the midpoints, G, H and I, the Höhenfußpunkte, J, K and L, the centers of the upper vertical portions of the S orthocenter.

Special cases

  • The Feuerbach circle is exactly then through a corner of the triangle (ie, the vertex of the right angle ) when the triangle is right-angled.
  • The Feuerbach circle touches just then a side of the triangle ( ie the base ) when the triangle is isosceles.
  • The Feuerbach circle is true if and only match the inscribed circle, if the triangle is equilateral.

Properties

  • The Feuerbach circle touches the incircle of the triangle and the three excircles including the triangle exclusive, this property is referred to as the set of Feuerbach. The point in which Feuerbach circle and inscribed circle touch is called the Feuerbach point of the triangle. (Caution: Some, mostly German, authors refer to the center of the Feuerbach circle as " Feuerbach point .")
  • The center of the Feuerbach circle lies exactly in the middle between the orthocenter and circumcenter, ie also on the Euler line.
  • The radius of the Feuerbach circle is half as large as the circumradius of the triangle.
  • The Feuerbach circle bisects the distance between the orthocenter and any point on the perimeter.
  • If an equilateral ( rectangular ) hyperbola through the vertices of a triangle, then its center lies on the Feuerbach circle.
  • The center of the Kiepert hyperbola lies on the Feuerbach circle.

Coordinates

History

Historically, it should be noted that the Feuerbach circle was discovered a year before the publication of the relevant font Feuerbach, so in 1821, by Charles Julien Brianchon and Jean Victor Poncelet. In Germany but the name Feuerbach circle has become the norm. The reason for this is that of Feuerbach derived, relatively difficult proof that this circle touches the incircle and the excircles. In the rest of the world they say mostly nine-point circle. It is also spreading the historically equitable designation Euler circuit.

Equilateral triangle Trilinear coordinates Max Koecher Triangle center
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