\documentclass{standalone}
\usepackage{tikz}

\begin{document}
\begin{tikzpicture}[scale=1]

\coordinate (O) at (0,0);
\draw (O) circle[radius=3cm];

\coordinate (A) at (0:3);
\coordinate (B) at (50:3);
\coordinate (C) at (140:3);
\coordinate (D) at (220:3);

\coordinate (Q) at (intersection of A--B and C--D);
\coordinate (R) at (intersection of A--D and B--C);
\coordinate (P) at (intersection of A--C and B--D);

% Draw the quadrilateral
\draw (A) -- (B) -- (C) -- (D) -- (A);

% Draw diagonals
\draw (A) -- (C);
\draw (B) -- (D);

% Draw dashed extensions for Q and R
\draw[dashed] (B) -- (Q);
\draw[dashed] (C) -- (Q);
\draw[dashed] (A) -- (R);
\draw[dashed] (B) -- (R);

% Points and labels
\fill (A) circle (1.5pt) node[below right] {A};
\fill (B) circle (1.5pt) node[above right] {B};
\fill (C) circle (1.5pt) node[above left] {C};
\fill (D) circle (1.5pt) node[below left] {D};
\fill (O) circle (1.5pt) node[below left] {O};
\fill (P) circle (1.5pt) node[below] {P};
\fill (Q) circle (1.5pt) node[above] {Q};
\fill (R) circle (1.5pt) node[right] {R};

% Draw the triangle PQR with dashed lines
\draw[dashed] (P) -- (Q) -- (R) -- (P);

\end{tikzpicture}
\end{document}