As shown in Figure 4-1, given that three points $A, B, C$ on the parabola $y=x^{2}$ form an isosceles right triangle with vertex $B$ being the right-angle vertex.
(1) If the slope of line $B C$ is $k$, find the coordinates of vertex $B$;
(2) When the slope $k$ of line $B C$ varies, find the minimum area of the isosceles right triangle $A B C$.
15. (1) Let the coordinates of points $A, B, C$ be $A\left(x_{1}, y_{1}\right), B\left(x_{2}, y_{2}\right), C\left(x_{3}, y_{3}\right)$. Since the slope of line $BC$ is $k$, the slope of line $AB$ is $-\frac{1}{k}$. Therefore, we have $y_{3}-y_{2}=k\left(x_{3}-x_{2}\right), y_{1}-y_{2}=-\frac{1}{k}\left(x_{1}-x_{2}\right)$.

Since points $A, B, C$ are on the parabola, we have $y_{1}=x_{1}^{2}, y_{2}=x_{2}^{2}, y_{3}=x_{3}^{2}$. Substituting these into the above equations, we get $x_{3}+x_{2}=k, x_{1}+x_{2}=-\frac{1}{k}$. Thus, $x_{3}=k-x_{2}, x_{1}=-\frac{1}{k}-x_{2}$.
Given that $|AB|=|BC|$, we have $\sqrt{\left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}}=\sqrt{\left(x_{3}-x_{2}\right)^{2}+\left(y_{3}-y_{2}\right)^{2}}$,

which simplifies to $\sqrt{1+\frac{1}{k^{2}}}\left(x_{2}-x_{1}\right)=\sqrt{1+k^{2}}\left(x_{3}-x_{2}\right), x_{2}-x_{1}=k\left(x_{3}-x_{2}\right)$.
Therefore, $x_{2}-\left(-\frac{1}{k}-x_{2}\right)=k\left(k-x_{2}-x_{2}\right)$, which simplifies to $x_{2}=\frac{k^{3}-1}{2 k(k+1)}$. Hence, the coordinates of point $B$ are $\left(\frac{k^{3}-1}{2 k(k+1)}, \frac{\left(k^{3}-1\right)^{2}}{4 k^{2}(k+1)^{2}}\right)$. Since $1 \geqslant 0$, we have $k \geqslant 1$.
The length of the legs of the isosceles right triangle is
$$
\begin{aligned}
\sqrt{k^{2}+1}\left(x_{1}-x_{2}\right) & =\sqrt{k^{2}+1}\left(k-2 x_{2}\right) \\
& =\sqrt{k^{2}+1}\left(k-\frac{k^{3}-1}{k(k+1)}\right) \\
& =\frac{k^{2}+1}{k} \cdot \frac{\sqrt{k^{2}+1}}{k+1} \\
& \geqslant \frac{2 k}{k} \cdot \frac{\sqrt{\frac{(k+1)^{2}}{2}}}{k+1} \\
& =\sqrt{2} .
\end{aligned}
$$

Equality holds if and only if $k=1$, i.e., when point $B$ is at the origin. Therefore, the minimum area of the isosceles right triangle is 1.