$19 \cdot 1$ If all the diagonals of a convex $n$-sided polygon $F (n \geqslant 4)$ are equal, then
(A) $F \in\{$ quadrilateral $\}$.
(B) $F \in\{$ pentagon $\}$.
(C) $F \in\{$ quadrilateral $\} \cup\{$ pentagon $\}$.
(D) $F \in\{$ equilateral polygon $\} \cup\{$ equiangular polygon $\}$.
(China High School Mathematics League, 1982)
[Solution] Using a regular pentagon as a counterexample can negate conclusion (A). Using a square as a counterexample can negate conclusion (B). Using an isosceles trapezoid as a counterexample can negate conclusion (D). Therefore, the correct conclusion can only be (C). Hence, the answer is $(C)$.