Question 6
Let $Sxyz$ be a trihedral angle with vertex $S$, where $\angle xSz=\alpha$, $\angle xSy=\beta$, $\angle ySz=\gamma$. Let $A,B,C$ be its dihedral angles at edges $Sy,Sz,Sx$, respectively.
(a) Prove that (i) $\sin\alpha/\sin A=\sin\beta/\sin B=\sin\gamma/\sin C$; (ii) $\alpha+\beta=90^\circ$ if and only if $A+B=180^\circ$.
(b) Suppose $\alpha=\beta=\gamma=90^\circ$. Fix $O$ on $Sz$ with $SO=a$. Points $M,N$ move on $Sx,Sy$ with $SM+SN=a$. Prove that $\angle SOM+\angle SON+\angle MON$ is constant. Find the locus of the center $I$ of the sphere circumscribed about the quadrilateral $OSMN$ (source wording).
(a) Prove that (i) $\sin\alpha/\sin A=\sin\beta/\sin B=\sin\gamma/\sin C$; (ii) $\alpha+\beta=90^\circ$ if and only if $A+B=180^\circ$.
(b) Suppose $\alpha=\beta=\gamma=90^\circ$. Fix $O$ on $Sz$ with $SO=a$. Points $M,N$ move on $Sx,Sy$ with $SM+SN=a$. Prove that $\angle SOM+\angle SON+\angle MON$ is constant. Find the locus of the center $I$ of the sphere circumscribed about the quadrilateral $OSMN$ (source wording).