Let $A = (-4,5,-17)$ and $B = (34,15,5).$  Let $P$ be the set of points in space such that triangle $ABP$ is equilateral.  The set of points $P$ in space that satisfy this condition traces a curve.  Find the total length of this curve.
Note that $AB = \sqrt{38^2 + 10^2 + 22^2} = 26 \sqrt{3}.$  Let $O$ be the midpoint of $\overline{AB}.$

[asy]
unitsize(1.5 cm);

pair A, B, P;

A = (-1,0);
B = (1,0);
P = (0,sqrt(3));

draw(A--B--P--cycle);
draw(yscale(sqrt(3))*xscale(0.4)*Circle((0,0),1),dashed);
draw(P--(A + B)/2);

label("$A$", A, W);
label("$B$", B, E);
label("$P$", P, N);

dot("$O$", (A + B)/2, S);
[/asy]

Then $AO = 13 \sqrt{3}.$  The set of points $P$ such that triangle $ABP$ is equilateral is a circle, centered at $O$ with radius
\[OP = AO \sqrt{3} = 39.\]The circumference of this circle is then $2 \pi \cdot 39 = \boxed{78 \pi}.$