MHB Prove Geometry Challenge: Cyclic Quadrilateral PQRS

[anemone]

Given a cyclic quadrilateral $PQRS$ where $PQ=p,\,QR=q,\,RS=r$, $\angle PQR=120^{\circ}$ and $\angle PQS=30^{\circ}$.

Prove that $|\sqrt{r+p}-\sqrt{r+q}|=\sqrt{r-p-q}$

 

[Albert1]

Given a cyclic quadrilateral $PQRS$ where $PQ=p,\,QR=q,\,RS=r$, $\angle PQR=120^{\circ}$ and $\angle PQS=30^{\circ}$.

Prove that $|\sqrt{r+p}-\sqrt{r+q}|=\sqrt{r-p-q}$

Spoiler

my solution :

View attachment 4127

 

[anemone]

Thank you Albert for your solution and I will post the answer that I have at hand later, in case there are others who might want to try it..