Triihuk

En triihuk as en geomeetrisk figuur. Hat hää trii sidjen ${\displaystyle a,b,c}$, diar oner trii winkel ${\displaystyle \alpha ,\beta ,\gamma }$ uun trii huken ${\displaystyle A,B,C}$ tuupluup.

Tekst üüb Öömrang

Algemian triihuk

Mä triihuken befaadet ham en aanj fääk uun a matematiik, det trigonometrii.

Bereegnangen

Triihuk mä gratin, diar uun det tabel näämd san.

Amkreisraadius:	${\displaystyle r={\frac {a}{2\sin \alpha }}={\frac {b}{2\sin \beta }}={\frac {c}{2\sin \gamma }}}$
Amfaadang:	${\displaystyle u=8r\cdot \cos {\frac {\alpha }{2}}\cdot \cos {\frac {\beta }{2}}\cdot \cos {\frac {\gamma }{2}}=a+b+c}$
Bankreisraadius:	${\displaystyle \rho =4r\cdot \sin {\frac {\alpha }{2}}\cdot \sin {\frac {\beta }{2}}\cdot \sin {\frac {\gamma }{2}}={\frac {2A}{u}}}$
Hööchde:	${\displaystyle h_{a}=c\cdot \sin \beta =b\cdot \sin \gamma }$
	${\displaystyle h_{b}=a\cdot \sin \gamma =c\cdot \sin \alpha }$
	${\displaystyle h_{c}=b\cdot \sin \alpha =a\cdot \sin \beta }$
Hööchde: (Bereegnang mä sidjenlengden)	${\displaystyle h_{a}={\sqrt {2(a^{2}b^{2}+b^{2}c^{2}+c^{2}a^{2})-(a^{4}+b^{4}+c^{4})}}/(2a)}$
	${\displaystyle h_{b}={\sqrt {2(a^{2}b^{2}+b^{2}c^{2}+c^{2}a^{2})-(a^{4}+b^{4}+c^{4})}}/(2b)}$
	${\displaystyle h_{c}={\sqrt {2(a^{2}b^{2}+b^{2}c^{2}+c^{2}a^{2})-(a^{4}+b^{4}+c^{4})}}/(2c)}$
Areal:	${\displaystyle A={\frac {1}{2}}ah_{a}={\frac {1}{2}}bh_{b}={\frac {1}{2}}ch_{c}}$
	${\displaystyle 16A^{2}=\left(a^{2}+b^{2}+c^{2}\right)^{2}-2\left(a^{4}+b^{4}+c^{4}\right)}$
	${\displaystyle 16A^{2}=4a^{2}c^{2}-\left(a^{2}+c^{2}-b^{2}\right)^{2}}$
	Heron sin formel: ${\displaystyle A={\sqrt {s(s-a)(s-b)(s-c)}}}$ mä: ${\displaystyle s={\frac {a+b+c}{2}}={\frac {u}{2}}}$
Swaarponkt: (geomeetrisk swaarponkt)	${\displaystyle x_{s}={\frac {1}{3}}(x_{A}+x_{B}+x_{C})}$
	${\displaystyle y_{s}={\frac {1}{3}}(y_{A}+y_{B}+y_{C})}$

Rochtwinkelt triihuk

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 ${\displaystyle h=h_{c}}$ faan en rochtwinkelt triihuk dialt det hüpotenuus iin uun tau dialen ${\displaystyle p}$ an ${\displaystyle q}$, an jo letj ufdiald triihuken mä a sidjen ${\displaystyle p}$, ${\displaystyle a}$, ${\displaystyle h}$ an ${\displaystyle q}$, ${\displaystyle h}$, ${\displaystyle b}$ san uk weder rochtwinkelt triihuken.

Wan dü tau faan jodiar sääks gratin (${\displaystyle a}$, ${\displaystyle b}$, ${\displaystyle c}$, ${\displaystyle p}$, ${\displaystyle q}$ an ${\displaystyle h}$) käänst, könst dü jo öödern bereegne efter:

Pythagoras sin reegel	${\displaystyle c^{2}=a^{2}+b^{2}}$
Euklid sin kateetenreegel	${\displaystyle a^{2}=c\cdot p}$
	${\displaystyle b^{2}=c\cdot q}$
Euklid sin hööchdereegel	${\displaystyle h^{2}=p\cdot q}$

Luke uk diar

• Trigonometrii

Wikiquote: Triihuken (sjiisk)

 Commonskategorii: Triihuken – Saamlang faan bilen of filmer

Wikibooks: Trigonometrii (sjiisk)
Wikibooks: Konstruktjuun faan triihuken (sjiisk)