Consider a $S$ sphere, planes $p_1,\dots,p_6$ and points $A,B,C,P,Q$ in Euclidean space.
We know that each of the six planes is tangent to the sphere. Planes $p_1,p_2,p_3$ have a common point, $P$, and the other three planes also have a common point: $Q$. Also, the intersection lines of planes $p_1-p_4, p_2-p_5, p_3-p_6$ are $AB, AC, BC$, respectively.
Suppose that $PQ$ meets $\triangle ABC$ at $R$. Also, let $C$ be the point on $S$, such that the distance between $C$ and plane $ABC$ is minimal.
Furthermore let $S’$ be the sphere tangent to planes $ABC,ABQ,ACQ,BCQ$. Prove that if $\triangle ABC$ and $S$ don’t intersect, then $CR$ is a diameter of $S’$.
I really don’t know how to prove it, I am very thankful for every solution. Sorry for not posting a figure, I don’t really like geogebra for 3d. If something is not clear, please comment!