Circle of Apollonios

In geometry , the circle of Apollonios (also circle of Apollonius or Apollonian circle ) is a special geometric place , namely the set of all points for which the ratio of the distances to two given points has a given value. The circle of Apollonios is not to be confused with the Apollonian problem , a circle of contact problem. The namesake in both cases is Apollonios von Perge .

Sentence and definition

Circle of Apollonios

• A segment and a positive real number are given . Then the set of points is a circle called the circle of Apollonios. ${\displaystyle [AB]}$ ${\displaystyle \lambda \neq 1}$
  ${\displaystyle k_{A}=\{X|{\overline {AX}}:{\overline {XB}}=\lambda \}}$
  

In support of the circular property using the inner and outer division point of the track in proportion . These two points ( and ) meet the condition required above and share the route harmoniously . If there is any point with the property , then the straight line divides the given segment proportionally . must therefore coincide with the bisector of the angle . Accordingly, it can be shown that the straight line bisects the secondary angle of . Since the bisectors of secondary angles are perpendicular to each other, must be on the Thales circle above . ${\displaystyle [AB]}$ ${\displaystyle \lambda }$ ${\displaystyle T_{i}}$ ${\displaystyle T_{a}}$ ${\displaystyle [AB]}$ ${\displaystyle X}$ ${\displaystyle {\overline {AX}}:{\overline {XB}}=\lambda }$ ${\displaystyle XT_{i}}$ ${\displaystyle [AB]}$ ${\displaystyle {\overline {XA}}:{\overline {XB}}}$ ${\displaystyle XT_{i}}$ ${\displaystyle AXB}$ ${\displaystyle XT_{a}}$ ${\displaystyle \angle AXB}$ ${\displaystyle X}$ ${\displaystyle [T_{i}T_{a}]}$

Conversely, every point of the Thales circle mentioned above fulfills the condition . ${\displaystyle X}$ ${\displaystyle {\overline {AX}}:{\overline {XB}}=\lambda }$

In the special case , the desired set of points is the mid-perpendicular of points A and B. ${\displaystyle \lambda =1}$

Other properties

• The radius of the circle of Apollonios is . ${\displaystyle r_{A}={\tfrac {\lambda }{|\lambda ^{2}-1|}}{\overline {AB}}}$

• The continuous circle of Apollonios for the route is the continuous inversion circle, related to which the end points are inversely to each other. ${\displaystyle T_{i}}$ ${\displaystyle [AB]}$ ${\displaystyle T_{i}}$ ${\displaystyle A,B}$

• If A and B merge into one another at inversion at the circle of Apollonios, each circle passing through A and B is also inverted in itself and therefore intersects the circle of Apollonios at right angles. This also applies in particular to the circle that is crossed. Because of the reciprocity of the harmonic division - one pair of points divides another harmoniously, so it is itself divided harmonically by this one (in proportion instead of ) - the circle is over the circle of Apollonios for the line . ${\displaystyle [AB]}$ ${\displaystyle {\tfrac {\lambda +1}{\lambda -1}}}$ ${\displaystyle \lambda }$ ${\displaystyle [AB]}$ ${\displaystyle [T_{i}T_{a}]}$

• The three circles of Apollonios of a triangle intersect at the isodynamic point of the corresponding triangle.

literature

• Franz Lemmermeyer: Mathematics à la carte: Quadratic equations with intersections of cones . Springer, 2016, ISBN 9783662503416 , p. 98
• Joachim Engel, Andreas Fest: Complex numbers and plane geometry . Walter de Gruyter, 2016, ISBN 9783110406887 , p. 40
• Nathan Altshiller: On the Circles of Apollonius . The American Mathematical Monthly, Vol. 22, No. 8 (Oct. 1915), pp. 261-263 ( JSTOR 2691113 )
• Roger A. Johnson: Advanced Euclidean Geometry . Dover 2007, ISBN 978-0-486-46237-0 , pp. 40, 294-297 (first published in 1929 by the Houghton Mifflin Company (Boston) under the title Modern Geometry ).

Web links

Commons : Apollonius circles  - collection of pictures, videos and audio files

• http://www.herder-oberschule.de/madincea/aufg0009/harmonie.pdf (PDF file; 261 kB)
• Apolloniuskreis on cut-the-knot.org
• David B. Surowski: Advanced High-School Mathematics - English script, p. 31