Formula from WKB Holz

The formula from WKB Holz , named after Walter KB Holz (1908–1993), is a mathematical formula that is located in the transition field between triangular geometry and circular geometry and with the help of which the radius of the inner Soddy circle of a triangle of the Euclidean plane can be calculated . The formula of wood is directly related to Descartes' theorem .

Representation of the formula

The formula says the following:

An arbitrary triangle of the Euclidean plane is given . ${\displaystyle ABC}$ ${\displaystyle \mathbb {R} ^{2}}$

As usual, the side lengths are denoted with , the halved circumference with , the incircle radius with and the three circle radii with . ${\displaystyle a,b,c}$ ${\displaystyle s}$ ${\displaystyle r}$ ${\displaystyle r_{a},r_{b},r_{c}}$

For let the respective circle around the corner point with the radius and let the inner Soddy circle for these three circles. ${\displaystyle (X,x)\in \{(A,a),(B,b),(C,c)\}}$ ${\displaystyle {\mathcal {K}}_{X}}$ ${\displaystyle X}$ ${\displaystyle s-x}$ ${\displaystyle {\mathcal {K}}}$

The radius of sei . ${\displaystyle {\mathcal {K}}}$ ${\displaystyle \sigma }$

Then the following equations apply :

(I) ${\displaystyle {\frac {1}{r}}={\frac {1}{r_{a}}}+{\frac {1}{r_{b}}}+{\frac {1}{r_{c}}}}$

(II) ${\displaystyle {\frac {1}{\sigma }}={\frac {1}{s-a}}+{\frac {1}{s-b}}+{\frac {1}{s-c}}+{\frac {2}{r}}}$

Explanations, notes and other formulas

• Here, a circle is always understood to be a circular line , that is to say a - dimensional compact subset of the Euclidean plane . According to this, a circle is to be distinguished from the circular disk belonging to it , i.e. from its convex envelope . ${\displaystyle 1}$ ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle {\mathcal {k}}\subset \mathbb {R} ^{2}}$ ${\displaystyle \operatorname {conv} ({\mathcal {k}})\subset \mathbb {R} ^{2}}$
• In addition to Frederick Soddy , Jakob Steiner and Ludwig Bieberbach also worked on Soddy circles and related questions about contact circles .
• According to HSM Coxeter , the circle described above always exists . Coxeter does not speak explicitly of the inner Soddy circle , but rather describes it. It is that circle of contact of the three circles , on the outside of which all three corner points of the triangle lie . So it can be said - and this is how Coxeter describes it - that the inner Soddy circle is that circle of contact between the three circles and that is enclosed by the three . ${\displaystyle {\mathcal {K}}}$ ${\displaystyle {\mathcal {K}}_{X}}$ ${\displaystyle \mathbb {R} ^{2}\setminus {\operatorname {conv} ({\mathcal {K}})}}$ ${\displaystyle {\mathcal {K}}_{X}}$
• Unlike the case of the inner-circle must Soddy an outer Berührkreis to the three circuits , that is one whose circular disk , all corners of the triangle and the three contains not exist in every case. According to Coxeter, this is especially true in the event that the triangle is very "blunt" . ${\displaystyle {\mathcal {K}}^{'}}$ ${\displaystyle {\mathcal {K}}_{X}}$ ${\displaystyle {\operatorname {conv} }({\mathcal {K}}^{'})}$ ${\displaystyle {\mathcal {K}}_{X}}$ ${\displaystyle ABC}$
• If is the center of , then the following applies with regard to its distances to the three corner points of the triangle: ${\displaystyle S}$ ${\displaystyle {\mathcal {K}}}$

(III) ${\displaystyle {\overline {SX}}=\sigma +s-x\;\;(\,(X,x)\in \{(A,a),(B,b),(C,c)\}\,)}$

• The following equation applies:

(IV) ${\displaystyle {\frac {2}{(s-a)^{2}}}+{\frac {2}{(s-b)^{2}}}+{\frac {2}{(s-c)^{2}}}+{\frac {2}{{\sigma }^{2}}}=({\frac {1}{s-a}}+{\frac {1}{s-b}}+{\frac {1}{s-c}}+{\frac {1}{\sigma }})^{2}}$

See also

• Apollonian problem
• Heron's formula

literature

• HSM Coxeter: Immortal Geometry . Translated into German by J. J. Burckhardt (=  Science and Culture . Volume 17 ). Birkhäuser Verlag, Basel / Stuttgart 1963 ( MR0692941 ).
• Nikolaos Dergiades: The Soddy circles . In: Forum Geometricorum . tape 7 , 2007, p. 191-197 ( MR2373402 ).

Individual evidence

1. ↑ HSM Coxeter: Immortal Geometry. 1963, p. 31.
2. ↑ ^{a b} Coxeter, op. Cit., Pp. 28-31.
3. ↑ Coxeter, op.cit, pp. 28, 30, 31.
4. ↑ Coxeter, op. Cit., Pp. 28-29.
5. ↑ Coxeter, op.cit, p. 29.
6. ↑ Coxeter, op.cit, p. 30.