1. As shown in Figure 4, an equilateral triangle \( \triangle ABC \) with side length 26 is inscribed in a circle, and chord \( DE \parallel BC \) intersects \( AB \) and \( AC \) at points \( F \) and \( G \), respectively. If the length of \( AF \) is \( x \) and the length of \( DF \) is \( y \), and both \( x \) and \( y \) are positive integers, then the value of \( y \) is ( ).
(A) 6
(B) 8
(C) 12
(D) 16
(7th Junior High School Mathematics Competition in Jiangsu Province)
Obviously, $D F=G E=y$.
Since $A F \cdot F B=D F \cdot F E$, therefore,
$$
x(26-x)=y(x+y) \text {. }
$$

Substituting the four options, only when $y=6$, $x$ meets the requirement.