While reading a proof in which they have defined the following homothety $$\begin{align*} h \colon C &\to C\\ x &\mapsto a+t(x-a) \end{align*}$$ where $a\in C$, C is a convex set of a normed space E and $t\in [0,1]$;
I didn't understand why they said that this homothety transform the ball $B(b,r)$ to $B(a+t(b-a),rt)$, where $b\in int(C)$. I tried to show that but in vain, any hint will be very helpful. Thank you for your time.
Hint: The map $h$ is a composition of three simpler maps.
The first map is a translation which shifts $a$ to the origin.
The second map is dilation by a factor of $t$.
The third map is a translation which shifts the origin back to $a$.
Now, determine what each of these simple maps does to the ball $B(b,r)$ as you apply the maps one by one.
Addendum: You may find the following useful. For $t\in (0,1]$, $$||x-b||<r \iff ||t(x-b)||<rt$$ and $$t(x-b)=a+t(x-a) -[a+t(b-a)] =h(x)-[a+t(b-a)] $$