We are given a circle, say $S$, and a point, say $P$, inside $S$. I want to find the radius of the smallest and the largest circles that pass through $P$ and touche $S$ (internally).
Intuitively it is clear that diameters of such circles can be chosen to be along the diameter of $S$ that passes through $P$. But I am unable to prove it.
I suppose there is a role of triangle inequality but I am not able see the useful triangles.
How can I prove this "obvious" fact?
As you suspected, the triangle inequality plays a role. As a rule of thumb, when there is a tangent point, we should draw the segment between the center of the circle and the tangent point.
Let $O$ be the center of $S$. Let $QR$ be the diameter of $S$ that pass through $P$, where $Q$ is nearer to $P$ than $R$. (Please check the special case when $P$ is $O$.)
Suppose another circle with center $X$ passes through $P$ and touches $S$ at $Y$, where $Y$ is neither $Q$ nor $R$ (Please check what happens if $Y$ is either $Q$ or $R$). Then $X$ is on the segment between $O$ and $Y$.
$$\begin{aligned} PQ&=OQ-OP=OY-OP\\ &< YP\\ &< XY+XP\end{aligned}$$ $$\begin{aligned} PR&=OR+OP=OY+OP\\ &=(XY+OX)+OP\\ &> XY+XP \end{aligned}$$
Since $XY+XP$ is two times the radius of the inner circle, we are done.
An exercise
What happens if $P$ is outside of $S$? Prove your conclusion.