Calculate area of a parallelogram given its sides and the angle between diagonals The parallelogram ABCD has sides AB = 80 cm and BC = 60 cm. Let X be the intersection of its diagonals. How to calculate the area of the parallelogram, when given the angle between diagonals BXC = 60°.
I have calculated the angle AXB = 120° and written two equations based on the cosine law, but it has started to be complicated and I hope there is more elegant way.
Picture of the Parallelogram with the given values
 A: You can use the law of cosines. You write the following system:
\begin{equation}
\begin{cases}
   80^2=x^2+y^2-2xy\cos120°\\60^2=x^2+y^2-2xy\cos60°
   \end{cases}
\end{equation}
where $x$=$AX$ and $y$=$BX$.
Subtracting the equations:
\begin{equation}
   80^2-60^2=x^2+y^2-2xy\cos120°-(x^2+y^2-2xy\cos60°)
\end{equation}
we have:
$2800=-2xy(\cos120°-\cos60°)$
$2800=-2xy(-\frac{1}{2}-\frac{1}{2})$
$2800=2xy$
so $xy=1400$
Now you can just calculate the area of the two triangles ABX and BCX using:
\begin{equation}
Area=\frac{1}{2}xy\sin\alpha, 
\end{equation}
where $\alpha$ is the angle between $x$ and $y$ (120° in ABX and 60° in BCX). Then it is easy.
A: Let us work in meters in order to work with small numbers.
Setting $a=XB=XD, b=XA=XC,$  the law of cosines gives: 


*

*In triangle ABX : 


$$\tag{1}a^2+b^2+ab=0.8^2  \ \ \text{because} \ \ \cos(2\pi/3)=-1/2.$$


*

*In triangle BCX : 


$$\tag{2}a^2+b^2-ab=0.6^2   \ \ \text{because} \ \ \cos(\pi/3)=1/2.$$
The by adding and subtracting (1) and (2), we obtain the following system:
$$\cases{a^2+b^2=0.5\\ab=0.14}.$$
By squaring both sides of the second equation, setting $A=a^2,B=b^2$, we obtain:
$$\cases{A+B=0.5\\AB=0.0196}.$$
Thus $A$ and $B$ are solutions of quadratic equation $X^2- 0.5 X + 0.0196=0$ that will not be difficult to solve: $A=0.0428768$ and $B=0.457123$. From which 
$a=\sqrt{A}=0.207067$ and $b=\sqrt{B}=0.676109$. It it the values (divided by 100) found through Wolfram, throwing away negatives ones.)
Then you proceed by computing areas of triangles XBC and XCD using formula: length of a side $\times$ length of another side $\times \sin$(angle) (angle between these sides).
