Aug 26, 2020 Thread starter Admin #1 anemone MHB POTW Director Staff member Feb 14, 2012 3,802 Let $p,\,q,\,r,\,s,\,t \in \mathbb {R_+}$ satisfying $p^2+pq+q^2=s^2\\ q^2+qr+r^2=t^2\\r^2+rp+p^2=s^2-st+t^2$ Prove that $\dfrac{s^2-st+t^2}{s^2t^2}=\dfrac{r^2}{q^2t^2}+\dfrac{p^2}{q^2s^2}-\dfrac{pr}{q^2ts}$

Let $p,\,q,\,r,\,s,\,t \in \mathbb {R_+}$ satisfying $p^2+pq+q^2=s^2\\ q^2+qr+r^2=t^2\\r^2+rp+p^2=s^2-st+t^2$ Prove that $\dfrac{s^2-st+t^2}{s^2t^2}=\dfrac{r^2}{q^2t^2}+\dfrac{p^2}{q^2s^2}-\dfrac{pr}{q^2ts}$