Yes (I assume $A$ has the induced topology). The point $V:=(0,1)$, common endpoint of all segments $S_r:=\{(rt,1-t): t\in[0,1]\}$, is either fixed by the continuous function $f:A\to A$, or it is mapped into some $S_r\setminus\{V\}$. But then $f $ has a fixed point on $S_r$. (For instance: let $P_r$ be the orthogonal projection on the line $l_r$ containing $S_r$: then $P_rf_{|S_r}$ has a fixed point that is also a fixed point of $f$.

Yes (I assume $A$ has the induced topology). The point $V:=(0,1)$, common endpoint of all segments $S_r:=\{(rt,1-t): t\in[0,1]\}$, is either fixed by the continuous function $f:A\to A$, or it is mapped into some $S_r\setminus\{V\}$. But then $f $ has a fixed point on $S_r$.