Reflecting Path in a Square

Square $ABCD$ has sides of length $4$. Points $P$ and $Q$ lie on
$\overline{AD}$ and $\overline{CD}$, respectively, with $AP=\frac{8}{5}$
and $DQ=\frac{10}{3}$. A path begins along the segment from $P$ to $Q$
and continues by reflecting against the sides of $ABCD$ (with congruent
incoming and outgoing angles). If the path hits a vertex of the square,
it terminates there; otherwise it continues forever. At which vertex
does the path terminate?

\begin{tikzpicture}[x=1.55cm,y=1.55cm,line cap=round,line join=round,
                    every node/.style={font=\normalsize}]
  \coordinate (A) at (0,0);
  \coordinate (B) at (4,0);
  \coordinate (C) at (4,4);
  \coordinate (D) at (0,4);
  \coordinate (P) at (0,{8/5});
  \coordinate (Q) at ({10/3},4);

  % The sides of the square.
  \draw[black,line width=0.8pt] (A)--(B)--(C)--(D)--cycle;

  % Equal incoming and outgoing angles at Q.
  \pgfmathsetmacro{\reflectionangle}{atan(18/25)}
  \draw[black,line width=0.7pt]
    (Q) ++(180:0.30) arc[start angle=180,
      end angle={180+\reflectionangle},radius=0.30];
  \draw[black,line width=0.7pt]
    (Q) ++({-\reflectionangle}:0.30)
      arc[start angle={-\reflectionangle},end angle=0,radius=0.30];

  % Initial segment and the beginning of the reflected path.
  \draw[red!85!black,line width=1.0pt] (P)--(Q);
  \draw[red!85!black,line width=1.0pt,
        -{Stealth[length=3.0mm,width=2.0mm]}]
    (Q)--({10/3+0.59},{4-(18/25)*0.59});

  % Points and labels.
  \foreach \point in {A,B,C,D,P,Q}
    \fill[black] (\point) circle[radius=1.25pt];
  \node[below left,inner sep=3pt] at (A) {$A$};
  \node[below right,inner sep=3pt] at (B) {$B$};
  \node[above right,inner sep=3pt] at (C) {$C$};
  \node[above left,inner sep=3pt] at (D) {$D$};
  \node[left,inner sep=4pt] at (P) {$P$};
  \node[above,inner sep=4pt] at (Q) {$Q$};
\end{tikzpicture}

(A) $A$ \qquad
(B) $B$ \qquad
(C) $C$ \qquad
(D) $D$ \qquad
(E) The path continues forever.

Solutions

• Solution 1