Apollonian circles

In geometry, the circle of Apollonius (also circle of Apollonius or apollonischer circle ) is a specific locus, namely the set of all points for which the ratio of the distances to two given points has a predetermined value. The circle of Apollonius is not to be confused with the Apollonian problem, a Berührkreis problem. It is named in both cases, Apollonius of Perga.

Sentence and definition

  • Given a line and a positive real number. Then the set of points a circle is called the circle of Apollonius.

In support of the county property using the inner and outer division point of the route in proportion. These two points (and ) meet the above required condition and share the route harmonious. Is now an arbitrary point with the property, then the straight line divides the given distance in the relationship. must therefore be consistent with the bisector of the angle. Accordingly, it can be shown that the straight angle of the side cut in half. Since the bisector of side angles are perpendicular to each other, must be on the Thales circle over.

Conversely fulfills each point of said Thales circle the condition.

In the specific case the requested set of points is the perpendicular bisector of the points A and B.

Other properties

  • Is the radius of the Apollonius circle.
  • The continuous Apollonioskreis for the route is on the end points are inverse to each other through inversion circuit based.
  • If A and B merge with inversion at Apollonioskreis in another, each passing through A and B circle is also inverted into itself and cuts the Apollonioskreis therefore perpendicular. This is particularly true for the crossed one circle. Due to the reciprocity of the harmonic division - shares a pair of points to another harmoniously, it is even divided harmonically by this ( in relation instead ) - is the circle on Apollonioskreis for the route.
  • The three circles of Apollonius of a triangle intersect at isodynamic point of the corresponding triangle.