# 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.

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