Question 83
Consider the Euclidean plane, the points $A = (0,0)$ and $B = (1,0)$ and the open half-strip
$$
S = \{(x, y) : 0 \lt x \lt 1, y \gt 0\}
$$
with width 1 and vertices $A$ and $B$.
Find all functions $f: S \to S$ satisfying the following conditions for all $P, Q \in S$:
(i) $f(f(P)) = P$
(ii) If $P, Q, A$ are collinear, then $f(P), f(Q)$ and $B$ are collinear.
(iii) If $f(P) = P$ and $f(Q) = Q$, then there is a circle containing $A, B, P$ and $Q$.
$$
S = \{(x, y) : 0 \lt x \lt 1, y \gt 0\}
$$
with width 1 and vertices $A$ and $B$.
Find all functions $f: S \to S$ satisfying the following conditions for all $P, Q \in S$:
(i) $f(f(P)) = P$
(ii) If $P, Q, A$ are collinear, then $f(P), f(Q)$ and $B$ are collinear.
(iii) If $f(P) = P$ and $f(Q) = Q$, then there is a circle containing $A, B, P$ and $Q$.