Circle

Triangle with circles (red)

The three excircles belong to the district and the inscribed circle to the special circles of a triangle that already in the ancient times were studied by Greek mathematicians.

The circles are defined as circles that are touched tangentially by one side of the triangle from the outside and by the extensions of the other two sides . Any triangle has three circles. The center points of the circle each lie on the bisector of an interior angle and on the bisector of the two exterior angles that do not belong to the interior angle.

Radii

The radius of the circle that touches the side ( ) inside results from ${\displaystyle a}$ ${\displaystyle [BC]}$

${\displaystyle \rho _{a}={\frac {A}{s-a}}}$,

it stands for the area and for half the circumference of the triangle . ${\displaystyle A}$ ${\displaystyle s}$ ${\displaystyle s={\tfrac {1}{2}}(a+b+c)}$

The radii and the two other circles are calculated in the same way . ${\displaystyle \rho _{b}}$ ${\displaystyle \rho _{c}}$

If one expresses the area according to Heron's theorem by the lengths of the sides, one obtains

${\displaystyle \rho _{a}={\sqrt {\frac {s(s-b)(s-c)}{s-a}}}}$.

The same applies to the other two arrivals

${\displaystyle \rho _{b}={\sqrt {\frac {s(s-a)(s-c)}{s-b}}}}$and . ${\displaystyle \rho _{c}={\sqrt {\frac {s(s-a)(s-b)}{s-c}}}}$

Contact point distances

Triangle, contact point distances of the circles, distances of the same color have the same lengths

designation

• ${\displaystyle c_{a}}$is the distance from to the points of contact of the circle with the side and with the extension of the side ${\displaystyle C}$ ${\displaystyle a}$ ${\displaystyle b.}$
• ${\displaystyle b_{a}}$is the distance from to the points of contact of the circle with the side and with the extension of the side ${\displaystyle B}$ ${\displaystyle a}$ ${\displaystyle c.}$

The index means that the circle is considered that touches the side in the triangle and not in the extension. The designation for the other two circles is chosen in the same way. ${\displaystyle a}$ ${\displaystyle a}$

The following applies:

${\displaystyle c_{a}=a_{c}=s-b}$

${\displaystyle c_{b}=b_{c}=s-a,}$

${\displaystyle a_{b}=b_{a}=s-c.}$

Here is half the circumference of the triangle. ${\displaystyle s}$

If you add a side length with a contact point distance of the circle on the side extension, this results ${\displaystyle s.}$

example

${\displaystyle c+b_{a}=b+c_{a}=s}$

Midpoints

The centers of the circles on their respective sides have the following barycentric coordinates, with the center of the circle representing side a: ${\displaystyle \displaystyle I_{a}}$

• ${\displaystyle \displaystyle I_{a}=(-a:b:c)}$
• ${\displaystyle \displaystyle I_{b}=(a:-b:c)}$
• ${\displaystyle \displaystyle I_{c}=(a:b:-c)}$

Construction of the circle centers

Triangle, construction of the circle centers

The following can be concluded from the introduction and the triangle with circles (red) above . The three center points of the circle can also be found solely by halving three outer angles which, as angle legs, each have one side and an extension of an adjacent side.

It starts with the extensions of the sides of the triangle beyond its corner points. Then follows z. As the bisector of the exterior angle at the apex of the angle leg side and extension of the side from the bisector of the external angle at the apex of the angle leg side and extension of the side from joins while delivering, as the intersection with the first Ankreismittelpunkt If all three Ankreismittelpunkte searched, the bisector of the outer angle at the apex with the angle legs side and extension of the side down is finally required. This results in the intersection points with the already existing bisector and also the two circle centers and ${\displaystyle ABC}$ ${\displaystyle w_{1}}$ ${\displaystyle C}$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle C.}$ ${\displaystyle w_{2}}$ ${\displaystyle B}$ ${\displaystyle a}$ ${\displaystyle c}$ ${\displaystyle B}$ ${\displaystyle w_{1}}$ ${\displaystyle I_{a}.}$ ${\displaystyle w_{3}}$ ${\displaystyle A}$ ${\displaystyle c}$ ${\displaystyle b}$ ${\displaystyle A}$ ${\displaystyle w_{1}}$ ${\displaystyle w_{2},}$ ${\displaystyle I_{b}}$ ${\displaystyle I_{c}.}$

Other properties

Triangle inscribed center ${\displaystyle I_{a}\;I_{b}\;I_{c},}$

• The circle centers and the triangle form a triangle whose height intersection is the inscribed center of the triangle . ${\displaystyle I_{a},I_{b}}$ ${\displaystyle I_{c}}$ ${\displaystyle ABC}$ ${\displaystyle H}$ ${\displaystyle ABC}$
• If you connect the corners of a triangle with the opposite contact points of the circles, the connecting straight lines intersect at one point, the nail point .

literature

• HSM Coxeter , SL Greitzer: Timeless geometry. Klett, Stuttgart 1983, ISBN 3-12-983390-0 .
• Max Koecher , Aloys Krieg : level geometry . 3. Edition. Springer-Verlag, Berlin 2007, ISBN 978-3-540-49327-3

Web links

Wiktionary: Ankreis  - explanations of meanings, word origins , synonyms, translations

• Eric W. Weisstein : Excircles . In: MathWorld (English).

Individual evidence

1. ↑ Wolf P. Barth: Geometry. 1.5 Circles at the triangle, Theorem 1.15 (circle radius). In: University of Magdeburg. Mathematical Institute of the University of Erlangen, August 10, 2004, p. 46 , accessed on September 1, 2019 . 
2. ↑ ^{a b} Wolf P. Barth: Geometry. 1.5 Circles at the triangle, Theorem 1.14 (auxiliary theorem). In: University of Magdeburg. Mathematical Institute of the University of Erlangen, August 10, 2004, p. 45 , accessed on August 31, 2019 .