En triihuk as en geomeetrisk figuur. Hat hää trii sidjen
a
,
b
,
c
{\displaystyle a,b,c}
, diar oner trii winkel
α
,
β
,
γ
{\displaystyle \alpha ,\beta ,\gamma }
uun trii huken
A
,
B
,
C
{\displaystyle A,B,C}
tuupluup.
Algemian triihuk
Mä triihuken befaadet ham en aanj fääk uun a matematiik , det trigonometrii .
Triihuk mä gratin, diar uun det tabel näämd san.
Amkreisraadius :
r
=
a
2
sin
α
=
b
2
sin
β
=
c
2
sin
γ
{\displaystyle r={\frac {a}{2\sin \alpha }}={\frac {b}{2\sin \beta }}={\frac {c}{2\sin \gamma }}}
Amfaadang:
u
=
8
r
⋅
cos
α
2
⋅
cos
β
2
⋅
cos
γ
2
=
a
+
b
+
c
{\displaystyle u=8r\cdot \cos {\frac {\alpha }{2}}\cdot \cos {\frac {\beta }{2}}\cdot \cos {\frac {\gamma }{2}}=a+b+c}
Bankreisraadius:
ρ
=
4
r
⋅
sin
α
2
⋅
sin
β
2
⋅
sin
γ
2
=
2
A
u
{\displaystyle \rho =4r\cdot \sin {\frac {\alpha }{2}}\cdot \sin {\frac {\beta }{2}}\cdot \sin {\frac {\gamma }{2}}={\frac {2A}{u}}}
Hööchde:
h
a
=
c
⋅
sin
β
=
b
⋅
sin
γ
{\displaystyle h_{a}=c\cdot \sin \beta =b\cdot \sin \gamma }
h
b
=
a
⋅
sin
γ
=
c
⋅
sin
α
{\displaystyle h_{b}=a\cdot \sin \gamma =c\cdot \sin \alpha }
h
c
=
b
⋅
sin
α
=
a
⋅
sin
β
{\displaystyle h_{c}=b\cdot \sin \alpha =a\cdot \sin \beta }
Hööchde: (Bereegnang mä sidjenlengden)
h
a
=
2
(
a
2
b
2
+
b
2
c
2
+
c
2
a
2
)
−
(
a
4
+
b
4
+
c
4
)
/
(
2
a
)
{\displaystyle h_{a}={\sqrt {2(a^{2}b^{2}+b^{2}c^{2}+c^{2}a^{2})-(a^{4}+b^{4}+c^{4})}}/(2a)}
h
b
=
2
(
a
2
b
2
+
b
2
c
2
+
c
2
a
2
)
−
(
a
4
+
b
4
+
c
4
)
/
(
2
b
)
{\displaystyle h_{b}={\sqrt {2(a^{2}b^{2}+b^{2}c^{2}+c^{2}a^{2})-(a^{4}+b^{4}+c^{4})}}/(2b)}
h
c
=
2
(
a
2
b
2
+
b
2
c
2
+
c
2
a
2
)
−
(
a
4
+
b
4
+
c
4
)
/
(
2
c
)
{\displaystyle h_{c}={\sqrt {2(a^{2}b^{2}+b^{2}c^{2}+c^{2}a^{2})-(a^{4}+b^{4}+c^{4})}}/(2c)}
Areal :
A
=
1
2
a
h
a
=
1
2
b
h
b
=
1
2
c
h
c
{\displaystyle A={\frac {1}{2}}ah_{a}={\frac {1}{2}}bh_{b}={\frac {1}{2}}ch_{c}}
16
A
2
=
(
a
2
+
b
2
+
c
2
)
2
−
2
(
a
4
+
b
4
+
c
4
)
{\displaystyle 16A^{2}=\left(a^{2}+b^{2}+c^{2}\right)^{2}-2\left(a^{4}+b^{4}+c^{4}\right)}
16
A
2
=
4
a
2
c
2
−
(
a
2
+
c
2
−
b
2
)
2
{\displaystyle 16A^{2}=4a^{2}c^{2}-\left(a^{2}+c^{2}-b^{2}\right)^{2}}
Heron sin formel:
A
=
s
(
s
−
a
)
(
s
−
b
)
(
s
−
c
)
{\displaystyle A={\sqrt {s(s-a)(s-b)(s-c)}}}
mä:
s
=
a
+
b
+
c
2
=
u
2
{\displaystyle s={\frac {a+b+c}{2}}={\frac {u}{2}}}
Swaarponkt : (geomeetrisk swaarponkt)
x
s
=
1
3
(
x
A
+
x
B
+
x
C
)
{\displaystyle x_{s}={\frac {1}{3}}(x_{A}+x_{B}+x_{C})}
y
s
=
1
3
(
y
A
+
y
B
+
y
C
)
{\displaystyle y_{s}={\frac {1}{3}}(y_{A}+y_{B}+y_{C})}
Rochtwinkelt triihuk mä di rocht winkel uun a ponkt C
Uun't hoonwerk hää'm det flooksis mä rochtwinkelt triihuken tu dun, det san triihuken mä en 90°-winkel . Det lung sidj jinauer di rocht winkel as det hüpotenuus , an jo kurt sidjen san a kateeten .
Det hööchde
h
=
h
c
{\displaystyle h=h_{c}}
faan en rochtwinkelt triihuk dialt det hüpotenuus iin uun tau dialen
p
{\displaystyle p}
an
q
{\displaystyle q}
, an jo letj ufdiald triihuken mä a sidjen
p
{\displaystyle p}
,
a
{\displaystyle a}
,
h
{\displaystyle h}
an
q
{\displaystyle q}
,
h
{\displaystyle h}
,
b
{\displaystyle b}
san uk weder rochtwinkelt triihuken.
Wan dü tau faan jodiar sääks gratin (
a
{\displaystyle a}
,
b
{\displaystyle b}
,
c
{\displaystyle c}
,
p
{\displaystyle p}
,
q
{\displaystyle q}
an
h
{\displaystyle h}
) käänst, könst dü jo öödern bereegne efter: