Triangle center

In the geometry is meant by the excellent points (also: strange points ) of a triangle in the first place the following four points:

  • The orthocenter H (intersection of heights),
  • The circumcenter U ( intersection of the perpendicular bisectors ( Seitensymmetralen ) )
  • The incenter I ( the intersection of the angle bisector ( Winkelsymmetralen )), and
  • The center of gravity S ( intersection of the medians (heavy lines)).

The orthocenter, the circumcenter and the centroid always lie on a straight line, the Euler's line. On it, in the middle between H and U, is also the center of Feuerbach circle.

More points after the Encyclopedia of Triangle Centers

In addition to the four "classic" excellent points of a triangle (focus, circumcenter, incenter, orthocenter ), which were already known in antiquity, were found and studied many other points in the last centuries. Clark Kimberling 's Encyclopedia of Triangle Centers ( see web link) leads to more than 5400 special points and their main characteristics. Introduced in this directory default name consisting of the letters X and an index that is used today in many treatises on the triangle geometry. The following table lists some important examples:

Related Topics

In addition to individual points can be a triangle also assign different tuple of points:

  • Morley triangle
  • Napoleon Triangle
  • Höhenfußpunktdreieck
  • Brocard points
  • Johnson Triangle
  • Kiepert Triangle

Special circuits are:

  • Circumference, inscribed circle, excircles
  • Feuerbach circle ( nine-point circle)
  • Lamoen Circle
  • Taylor Circle
  • Johnson Circles