Question 83
A sphere $\omega$ passes through a vertex $S$ of a triangular pyramid $SABC$ and intersects the edges $SA$, $SB$, $AC$ at $A_1 \neq S$, $B_1 \neq S$, and $C_1 \neq S$, respectively. The sphere $\omega$ intersects the circumsphere $\Omega$ of $SABC$ by a circle lying in a plane parallel to $(ABC)$. The points $A_2$, $B_2$, and $C_2$ are symmetric to the points $A_1$, $B_1$, and $C_1$ with respect to the midpoints of the segments $SA$, $SB$, $AC$, respectively. Prove that the points $A$, $B$, $C$, $A_2$, $B_2$, and $C_2$ lie on a sphere. (I. Bogdanov)