4.7 Delambre and Briggs law (for plane geometry)

  Topography notes

 

Triangle_with_notations_2

Delambre law

Similary to the Law of tangents, this law is a relation for the sides and the angles of a triangle.

\dfrac{a+b}{c}=\dfrac{\cos{(\frac{\alpha - \beta}{2})}}{\sin{(\gamma / 2)}} \dfrac{a-b}{c}=\dfrac{\sin{(\frac{\alpha - \beta}{2})}}{\cos{(\gamma / 2)}}
\dfrac{b+c}{a}=\dfrac{\cos{(\frac{\beta-\gamma}{2})}}{\sin{(\alpha / 2)}} \dfrac{b-c}{a}=\dfrac{\sin{(\frac{\beta-\gamma}{2})}}{\cos{(\alpha / 2)}}
\dfrac{c+a}{b}=\dfrac{\cos{(\frac{\gamma - \alpha}{2})}}{\sin{(\beta / 2)}} \dfrac{c-a}{b}=\dfrac{\sin{(\frac{\gamma - \alpha}{2})}}{\cos{(\beta / 2)}}

Briggs law

The briggs law formulas give an angle of a triangle knowing the sides, it’s used for determine an angle with one single step.

We call p=\frac{1}{2}(a+b+c) the semiperimeter of the triangle ABC.

\sin{\dfrac{\alpha}{2}}=\sqrt{\dfrac{(p-b)(p-c)}{bc}} \cos{\dfrac{\alpha}{2}}=\sqrt{\dfrac{p(p-a)}{bc}}  \tan{\dfrac{\alpha}{2}}=\sqrt{\dfrac{(p-b)(p-c)}{p(p-a)}}
 \sin{\dfrac{\beta}{2}}=\sqrt{\dfrac{(p-c)(p-a)}{ac}} \cos{\dfrac{\beta}{2}}=\sqrt{\dfrac{p(p-b)}{ac}} \tan{\dfrac{\beta}{2}}=\sqrt{\dfrac{(p-c)(p-a)}{p(p-b)}}
 \sin{\dfrac{\gamma}{2}}=\sqrt{\dfrac{(p-a)(p-b)}{ab}} \cos{\dfrac{\gamma}{2}}=\sqrt{\dfrac{p(p-c)}{ab}} \tan{\dfrac{\gamma}{2}}=\sqrt{\dfrac{(p-a)(p-b)}{p(p-c)}}

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