I'm currently trying to find the values of two angles (a and b) in this quadrilateral, where I know the diagonal (dashed) length and the lengths of two of the sides. These two side lengths are equal in length to one another and the other diagonal:
So far I've been able to express the values of the sides opposite to a and b in terms of the known side lengths and the angles, but I don't know how I can relate this to the (dashed) diagonal length, because this quadrilateral will not always be a parallelogram or a kite. Furthermore, the two diagonals wouldn't always intersect at their midpoints.
Is there any way that I can calculate the values of a and b using the values that I already have?
As stated the problem can have many solutions.
For example:
Let the quadrilateral $ABCD$ such that $AD=DB=BC=x$ and $AC=\sqrt{3}x$. See figure 1.
Figure 1
The angle $b$ depends on $a$ as the expression:
$$b = \frac{a}{2}- \frac {\pi}{2}+ \arccos(- \frac{1}{2 \sin {\frac{a}{2}}}+ \sin {\frac{a}{2}})$$
So we have many solutions, for example: $a=\frac{\pi}{3}$ and $b=\frac{\pi}{3}$ or $a=\frac{\pi}{4}$ and $b=\frac{\pi}{2}$ as shown in figure 2.
Figure 2