Question 31
Let $SH$ be an altitude in a tetrahedron $SABC$ and $H$ be inside the base $ABC$. A point $O$ on $SH$ is chosen so that $\angle AOS + \alpha = \angle BOS + \beta = \angle COS + \gamma = 180^\circ$, where $\alpha, \beta, \gamma$ are the dihedral angles corresponding to the edges $BC, AC, AB$ respectively. Let $A_1, B_1, C_1$ be the points of intersection of the following lines and planes: $A_1 = AO \cap SBC$, $B_1 = BO \cap SAC$, $C_1 = CO \cap SBA$. If the planes $ABC$ and $A_1B_1C_1$ are parallel to each other, prove that $SA = SB = SC$.
