Koordinatenwerte ---> Richtungsvektoren

${\displaystyle \mathbf {a} =\mathbf {p} _{3}-\mathbf {p} _{1}}$
${\displaystyle \mathbf {b} =\mathbf {p} _{2}-\mathbf {p} _{1}}$
${\displaystyle \mathbf {c} =\mathbf {b} -\mathbf {a} }$

Seitenlängen und Umfang

${\displaystyle a_{norm}=|\mathbf {a} |}$
${\displaystyle b_{norm}=|\mathbf {b} |}$
${\displaystyle c_{norm}=|\mathbf {c} |}$
${\displaystyle U\ =\ a_{norm}+b_{norm}+c_{norm}}$

Bedingung: ${\displaystyle a_{norm}\neq 0}$, ${\displaystyle b_{norm}\neq 0}$, ${\displaystyle c_{norm}\neq 0}$

Winkel

${\displaystyle \angle ab=\arccos {\frac {\mathbf {a} \cdot \mathbf {b} }{a_{norm}b_{norm}}}}$
${\displaystyle \angle bc=\arccos {\frac {\mathbf {b} \cdot \mathbf {c} }{b_{norm}c_{norm}}}}$
${\displaystyle \angle ac=\pi -\angle ab-\angle bc}$

Bedingung: ${\displaystyle \angle ab\neq 0}$, ${\displaystyle \angle ac\neq 0}$, ${\displaystyle \angle bc\neq 0}$

Fläche

Es gilt

${\displaystyle A_{Parallelogramm}=|\mathbf {a} \times \mathbf {b} |=a_{norm}b_{norm}\sin \angle ab}$

und somit

${\displaystyle A={\frac {A_{parallelogramm}}{2}}={\frac {a_{norm}b_{norm}\sin \angle ab}{2}}}$

Umkreis

Normalen:

${\displaystyle \mathbf {a} \cdot \mathbf {n} =0}$
${\displaystyle \mathbf {b} \cdot \mathbf {m} =0}$
${\displaystyle |\mathbf {n} |=1}$
${\displaystyle |\mathbf {m} |=1}$

${\displaystyle \Rightarrow }$

${\displaystyle n_{1}=-a_{2}{\sqrt {\frac {1}{a_{1}^{2}+a_{2}^{2}}}}}$
${\displaystyle n_{2}=a_{1}{\sqrt {\frac {1}{a_{1}^{2}+a_{2}^{2}}}}}$
${\displaystyle m_{1}=-b_{2}{\sqrt {\frac {1}{b_{1}^{2}+b_{2}^{2}}}}}$
${\displaystyle m_{2}=b_{1}{\sqrt {\frac {1}{b_{1}^{2}+b_{2}^{2}}}}}$

Geradenschnittpunkt:

${\displaystyle g1:\ \mathbf {x} _{1}={\frac {\mathbf {a} }{2}}+t_{1}\mathbf {n} }$
${\displaystyle g2:\ \mathbf {x} _{2}={\frac {\mathbf {b} }{2}}+t_{2}\mathbf {m} }$
Der Umkreismittelpunkt ergibt sich als Schnittpunkt dieser beiden Geraden: ${\displaystyle \mathbf {x} =\mathbf {x} _{1}=\mathbf {x} _{2}}$

${\displaystyle \Rightarrow }$

${\displaystyle t_{2}={\frac {n_{1}\left(a_{2}-b_{2}\right)+n_{2}\left(b_{1}-a_{1}\right)}{2\left(m_{2}n_{1}-m_{1}n_{2}\right)}}}$

Bedingung: ${\displaystyle m_{2}n_{1}-m_{1}n_{2}\neq 0}$

Umkreismittelpunkt und -radius:

${\displaystyle \mathbf {x} _{o}=\mathbf {p} _{1}+{\frac {\mathbf {b} }{2}}+t_{2}\mathbf {m} }$

${\displaystyle r_{o}=\left|{\frac {\mathbf {b} }{2}}+t_{2}\mathbf {m} \right|}$

Inkreis

${\displaystyle \mathbf {v} ={\frac {1}{2}}\left({\frac {\mathbf {a} }{a_{norm}}}+{\frac {\mathbf {b} }{b_{norm}}}\right)}$

${\displaystyle \mathbf {w} ={\frac {1}{2}}\left({\frac {\mathbf {b} }{b_{norm}}}+{\frac {\mathbf {c} }{c_{norm}}}\right)}$

${\displaystyle g1:\ \mathbf {x} _{1}=t_{1}\mathbf {v} }$

${\displaystyle g2:\ \mathbf {x} _{2}=\mathbf {b} +t_{2}\mathbf {v} }$

Der Inkreismittelpunkt ergibt sich als Schnittpunkt dieser beiden Geraden: ${\displaystyle \mathbf {x} =\mathbf {x} _{1}=\mathbf {x} _{2}}$

${\displaystyle \Rightarrow }$

${\displaystyle t_{2}={\frac {b_{1}v_{2}-b_{2}v_{1}}{w_{2}v_{1}-w_{1}v_{2}}}}$

Bedingung: ${\displaystyle w_{2}v_{1}-w_{1}v_{2}\neq 0}$

Inkreismittelpunkt:

${\displaystyle \mathbf {x} _{i}=\mathbf {p} _{2}+t_{2}\mathbf {w} }$

Inkreisradius:

${\displaystyle r_{i}={\frac {2A}{a_{norm}+b_{norm}+c_{norm}}}}$

Schwerpunkt

${\displaystyle \mathbf {v} =\mathbf {a} +{\frac {\mathbf {c} }{2}}}$
${\displaystyle \mathbf {w} ={\frac {\mathbf {b} }{2}}-\mathbf {a} }$

${\displaystyle g1:\ \mathbf {x} _{1}=t_{1}\mathbf {v} }$
${\displaystyle g2:\ \mathbf {x} _{2}=\mathbf {a} +t_{2}\mathbf {w} }$

Der Dreiecksschwerpunkt ergibt sich als Schnittpunkt dieser beiden Geraden: ${\displaystyle \mathbf {x} =\mathbf {x} _{1}=\mathbf {x} _{2}}$

${\displaystyle t_{2}={\frac {a_{1}v_{2}-a_{2}v_{1}}{w_{2}v_{1}-w_{1}v_{2}}}}$

Bedingung: ${\displaystyle w_{2}v_{1}-w_{1}v_{2}\neq 0}$

${\displaystyle \Rightarrow }$

${\displaystyle \mathbf {x} =\mathbf {a} +t_{2}\mathbf {w} }$

${\displaystyle \mathbf {x} _{cog}=\mathbf {p} _{1}+\mathbf {x} }$