Soddy Circle

Frederick Soddy

The Soddy circles are the solutions for a special case of the Apollonian problem , where the three given circles , whose centers are the corners of a triangle , touch each other. They are named after Frederick Soddy , who rediscovered Descartes' theorem based on these circles and published it on June 20, 1936 in the journal Nature in the form of a poem entitled The kiss precise .

definition

Given a triangle and the three circles with the centers , respectively , each defined by the contact points of the inscribed circle go with the adjacent sides of the triangle. (These three circles touch each other in pairs.) The two Soddy circles are now those circles which touch the three circles mentioned. In general, one differentiates between the inner and outer Soddy circles. ${\displaystyle ABC}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle C}$

properties

• According to Descartes' theorem, the following applies to the curvature of the two Soddy circles: ${\displaystyle k}$

${\displaystyle k=k_{1}+k_{2}+k_{3}\pm 2{\sqrt {k_{1}k_{2}+k_{2}k_{3}+k_{3}k_{1}}}.}$

In this case denote the curvatures (= reciprocals of the radii ) of the circles around the corner points A, B and C. ${\displaystyle k_{1},k_{2},k_{3}}$

• The center of the outer Soddy circle is the isoperimetric point .
• The center of the inner Soddy circle is the point of the same detour .
• The following applies to the radii of the Soddy circles

${\displaystyle r=\left|{\frac {\Delta }{4R+r\pm U}}\right|.}$

Here referred to the area of , the Inkreisradius , the radius radius and the circumference. The plus sign applies to the inner Soddy circle, the minus sign to the outer one. ${\displaystyle \Delta }$ ${\displaystyle ABC}$ ${\displaystyle r}$ ${\displaystyle R}$ ${\displaystyle U}$

• The radius of the inner Soddy circle is calculated using the formula from WKB Holz .
• The circles around the corners of the triangle are touched by the outer Soddy circle for including, for excluding. In the limit case ( ) there is an infinite radius, i.e. H. the outer Soddy circle becomes a common tangent. ${\displaystyle 4R+r<U}$ ${\displaystyle 4R+r>U}$ ${\displaystyle 4R+r=U}$

swell

• N. Dergiades: The Soddy Circles. In: Forum Geometricorum . Volume 7, 2007, pp. 191-197. ( Online version )
• Eric W. Weisstein : Soddy Circles . In: MathWorld (English).
• Soddy Circles on cut-the-knot

Web links

• Arthur Baragarm, Alex Kontorovich: Efficiently Constructing Tangent Circles - simple construction of the two Soddy circles
• isoperimetric and equal detour points - interactive illustration on Geogebratube