Area of a (planar) quadrilateral using two opposite sides and four angles
I guessed the following area formula for a (planar) quadrilateral using two opposite sides and four angles:(When $a=AB$ and $c=CD$,)
$S = \frac{a^2}{2(\cot A + \cot B)} + \frac{c^2}{2(\cot C + \cot D)}$
This formula can be applied to the quadrilateral satisfying $A+B \ne \pi$ and $C+D \ne \pi$, including convex, concave, and even self-intersecting cases.
I'd like to know the applicable case, inapplicable case and reasons for the validity of those cases. Are there any known papers or more elegant ways to derive this formula?
I guess there are many ways to explain those reasons. It greatly resembles $S = \frac{1}{2}(ab \sin B + cd \sin D)$. It seems that $b$ and $d$ can be determined using $a, c, A, B, C, D$.
1 answer
First, you seem to be using the same letters to signify both corner points and the corresponding angles, which to me is nonstandard. Not a serious issue.
Since you guessed the formula, I would start by checking that it works in various cases, and in particular checking that it gives the same outcome as known formulae for particular quadrilaterals; that is, that you can derive this formula from other known formulae and the other way around in the cases where both apply. Especially the latter activity is likely to lead to more insight on the formulae.
If everything points out to the formula working, you should try to prove it. Study how the formulae for the area of other geometric shapes is proven to learn more about how this can be done, if in doubt.
Good luck!
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