Solution
= Solution
Write $r_x=d(x,a)$. The prescribed function is symmetric and nonnegative. If $x\ne y$, both $r_x,r_y$ could be zero only if $x=y=a$, so $d_a(x,y)>0$. For the triangle inequality, if any two of $x,y,z$ coincide it is immediate. If all three are distinct,
$$
d_a(x,z)=r_x+r_z\le(r_x+r_y)+(r_y+r_z)=d_a(x,y)+d_a(y,z).
$$
Thus $\boxed{d_a\text{ is a metric}}$. Also $d_a(a,x)=d(a,x)$, a fact useful in the following parts.