Let $ABC$ be a triangle.  There exists a positive real number $k$, such that if the altitudes of triangle $ABC$ are extended past $A$, $B$, and $C$, to $A'$, $B'$, and $C'$, as shown, such that $AA' = kBC$, $BB' = kAC$, and $CC' = kAB$, then triangle $A'B'C'$ is equilateral.

[asy]
unitsize(0.6 cm);

pair[] A, B, C;
pair D, E, F;

A[0] = (2,4);
B[0] = (0,1);
C[0] = (5,0);
D = (A[0] + reflect(B[0],C[0])*(A[0]))/2;
E = (B[0] + reflect(C[0],A[0])*(B[0]))/2;
F = (C[0] + reflect(A[0],B[0])*(C[0]))/2;
A[1] = A[0] + (1/sqrt(3))*(rotate(90)*(C[0] - B[0]));
B[1] = B[0] + (1/sqrt(3))*(rotate(90)*(A[0] - C[0]));
C[1] = C[0] + (1/sqrt(3))*(rotate(90)*(B[0] - A[0]));

draw(A[0]--B[0]--C[0]--cycle);
draw(A[1]--D);
draw(B[1]--E);
draw(C[1]--F);

label("$A$", A[0], NW);
dot("$A'$", A[1], N);
label("$B$", B[0], S);
dot("$B'$", B[1], SW);
label("$C$", C[0], S);
dot("$C'$", C[1], SE);
[/asy]

Find $k$.
We place the diagram in the complex plane, so that the vertices $A$, $A'$, $B$, $B'$, $C$, and $C'$ go to the complex numbers $a$, $a'$, $b$, $b'$, $c$, and $c'$, respectively.

To get to $a'$, we rotate the line segment joining $b$ to $c$ by $90^\circ$ (which we achieve by multiplying $c - b$ by $i$).  Also, we want $AA' = kBC$, so we multiply this complex number by $k$ as well.  Hence,
\[a' = a + ki(c - b).\]Similarly,
\begin{align*}
b' &= b + ki(a - c), \\
c' &= c + ki(b - a).
\end{align*}[asy]
unitsize(0.6 cm);

pair[] A, B, C;
pair D, E, F;

A[0] = (2,4);
B[0] = (0,1);
C[0] = (5,0);
D = (A[0] + reflect(B[0],C[0])*(A[0]))/2;
E = (B[0] + reflect(C[0],A[0])*(B[0]))/2;
F = (C[0] + reflect(A[0],B[0])*(C[0]))/2;
A[1] = A[0] + (1/sqrt(3))*(rotate(90)*(C[0] - B[0]));
B[1] = B[0] + (1/sqrt(3))*(rotate(90)*(A[0] - C[0]));
C[1] = C[0] + (1/sqrt(3))*(rotate(90)*(B[0] - A[0]));

draw(A[0]--B[0]--C[0]--cycle);
draw(A[1]--D);
draw(B[1]--E);
draw(C[1]--F);
draw(B[1]--A[1]--C[1],dashed);

label("$a$", A[0], NW);
dot("$a'$", A[1], N);
label("$b$", B[0], S);
dot("$b'$", B[1], SW);
label("$c$", C[0], S);
dot("$c'$", C[1], SE);
[/asy]

We want triangle $A'B'C'$ to be equilateral, so we want $a'$, $b'$, and $c'$ to satisfy
\[c' - a' = e^{\pi i/3} (b' - a').\]Substituting our expressions for $a'$, $b'$, and $c'$, and using the fact that
\[e^{\pi i/3} = \frac{1}{2} + \frac{\sqrt{3}}{2} i,\]we get
\[c + ki(b - a) - a - ki(c - b) = \left( \frac{1}{2} + \frac{\sqrt{3}}{2} i \right) [b + ki(a - c) - a - ki(c - b)].\]Expanding and simplifying both sides, we get
\begin{align*}
&(-1 - ki) a + 2ki b + (1 - ki) c \\
&= \frac{-k \sqrt{3} - 1 + ki - i \sqrt{3}}{2} \cdot a + \frac{- k \sqrt{3} + 1 + ki + i \sqrt{3}}{2} \cdot b + (k \sqrt{3} - ki) c.
\end{align*}We want the coefficients of $a$, $b$, and $c$ to be equal on both sides.  Equating the coefficients of $c$, we get
\[1 - ki = k \sqrt{3} - ki,\]so $k = 1/\sqrt{3}$.  For this value of $k$, both coefficients of $a$ become $-1 - i/\sqrt{3}$, and both coefficients of $b$ become $2i/\sqrt{3}$.

Hence, the value of $k$ that works is $k = \boxed{\frac{1}{\sqrt{3}}}$.