Triangle $ABC$ has a right angle at $B$, and contains a point $P$ for which $PA = 10$, $PB = 6$, and $\angle APB = \angle BPC = \angle CPA$. Find $PC$.

[asy]
unitsize(0.2 cm);

pair A, B, C, P;

A = (0,14);
B = (0,0);
C = (21*sqrt(3),0);
P = intersectionpoint(arc(B,6,0,180),arc(C,33,0,180));

draw(A--B--C--cycle);
draw(A--P);
draw(B--P);
draw(C--P);

label("$A$", A, NW);
label("$B$", B, SW);
label("$C$", C, SE);
label("$P$", P, NE);
[/asy]
Since $\angle APB = \angle BPC = \angle CPA,$ they are all equal to $120^\circ.$

Let $z = PC.$  By the Law of Cosines on triangles $BPC,$ $APB,$ and $APC,$
\begin{align*}
BC^2 &= z^2 + 6z + 36, \\
AB^2 &= 196, \\
AC^2 &= z^2 + 10z + 100.
\end{align*}By the Pythagorean Theorem, $AB^2 + BC^2 = AC^2,$ so
\[196 + z^2 + 6z + 36 = z^2 + 10z + 100.\]Solving, we find $z = \boxed{33}.$
Since $\angle APB = \angle BPC = \angle CPA,$ they are all equal to $120^\circ.$

Let $z = PC.$  By the Law of Cosines on triangles $BPC,$ $APB,$ and $APC,$
\begin{align*}
BC^2 &= z^2 + 6z + 36, \\
AB^2 &= 196, \\
AC^2 &= z^2 + 10z + 100.
\end{align*}By the Pythagorean Theorem, $AB^2 + BC^2 = AC^2,$ so
\[196 + z^2 + 6z + 36 = z^2 + 10z + 100.\]Solving, we find $z = \boxed{33}.$
The final answer is $$\boxed{33}$$. I hope it is correct.