= Solution
Nonnegativity and symmetry follow from $d$, and $d_a(x,y)=0$ with $x\ne y$ would force both $x=a$ and $y=a$, a contradiction. For the <triangle inequality>, when the intermediate point $z$ is different from both endpoints,
$$
d_a(x,z)+d_a(z,y)=d(x,a)+2d(z,a)+d(a,y)\geq d_a(x,y).
$$
If $z$ equals an endpoint the inequality is immediate, and if $x=y$ its left target is zero. These cover every case, so \b[$d_a$ is a <metric>].
This is the <star metric centered at a point>.
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