It is not true. The area of the trapezoid is $a(b+c)/2$. Take the trapezoid $(-2,0), (2,0), (1,1), (-1,1)$. This has area $3$ and diagonal $\sqrt{10}$. If $a=b+c$, the area would be $\frac{a^2}{2}=5$ instead of $3$