Say we have a quadrilateral satisfying the constraint \begin{align*} (\vec{AD} + \vec{BC})\cdot \vec{AC} \geq 0, \quad (\vec{AD} + \vec{BC})\cdot \vec{BD} \geq 0,\\ (\vec{AB} + \vec{DC})\cdot \vec{DB} \geq 0, \quad (\vec{AB} + \vec{DC})\cdot \vec{AC} \geq 0. \end{align*} I wonder how severe this constraint is. Can it be translated to some other more intuitive constraints? (Especially angle constraints)
I have tried
\begin{align}
(\vec{AD} + \vec{BC})\cdot \vec{AC} = |AC|(|AD|\cos\angle DAC + |BC|\cos\angle ACB) = |AC|^2\left(\frac{\cos\angle DAC \sin\angle ACD}{\sin \angle ADC} + \frac{\cos\angle ACB \sin\angle BAC}{\sin\angle ABC}\right),
\end{align}
where the law of sines is used in the last equality. But it seems not help.
This is the quadrilateral:
