Preliminaries

Here several interesting and useful facts about triangles. Let $s$ denote the semiperimeter of triangle $ABC$ , $\alpha ,\beta, \gamma$ the angles, $a,b,c$ the opposite sides, and $K$ the area of the triangle.

1. $K = \frac {1}{2}ab\sin\gamma =\frac {1}{2}ac\sin\beta =\frac {1}{2}bc\sin\alpha.$
2. $K = \sqrt{s(s-a)(s-b)(s-c)}.$ (Heron's formula)
3. $c^2=a^2+b^2-2ab \cos(\gamma)$ (Law of Cosines).
4. $$2R = \frac{a}{\sin\alpha}=\frac{b}{\sin\beta}=\frac{c}{\sin\gamma}$$ (Law of Sines). ($R$ is the radius of the circumcircle)

See [2] for an elementary algebra proof of Heron's formula.
Let $s$ denote the semiperimeter of quadrilateral $ABCD$,
$\alpha ,\beta,\gamma, \delta$ the angles, $AB=a,BC=b,CD=c,DA=d$ and $Q$ the area.

1. $Q= \frac {1}{2}(a)(b)\sin\beta +\frac {1}{2}(d)(c)\sin\delta$.
2. $Q= \sqrt{(s-a)(s-b)(s-c)(s-d)-\frac {1}{2}abcd(\cos(\beta+\delta)+1)}$