Editing Ptolemy's theorem (section)

===Corollary 2. The law of cosines===
[[File:Ptolemy theore trig4.svg|thumb|Corollary 2: the law of cosines]]

Let <math>\theta_2=\theta_4</math>. The rectangle of corollary 1 is now a symmetrical trapezium with equal diagonals and a pair of equal sides. The parallel sides differ in length by <math>2x</math> units where:
:<math>x=S_2\cos(\theta_2+\theta_3)</math>

It will be easier in this case to revert to the standard statement of Ptolemy's theorem:

:<math>\begin{array}{lcl}
S_1 S_3 + S_2 S_4={\overline{AC}}\cdot{\overline{BD}}\\
\Rightarrow S_1 S_3+{S_2}^2={\overline{AC}}^2\\
\Rightarrow S_1[S_1-2S_2\cos(\theta_2+\theta_3)]+{S_2}^2={\overline{AC}}^2\\
\Rightarrow {S_1}^2+{S_2}^2-2S_1 S_2\cos(\theta_2+\theta_3)={\overline{AC}}^2\\
\end{array}</math>

The cosine rule for triangle ABC.