Three. (25 points) Given the quadratic function $y=a x^{2}+b x+c$ intersects the $x$-axis at points $A$ and $B$, and its vertex is $C$.
(1) Prove: If $\triangle A B C$ is an isosceles right triangle, then
$$
\Delta=b^{2}-4 a c=4 \text {; }
$$
(2) The quadratic function $y=x^{2}-(2 a+2) x+a^{2}+5 a+3$ intersects the $x$-axis at points $A$ and $B$, and intersects the line $y=3 x-1$ at point $C$. If $\triangle A B C$ is a right triangle, find the value of $a$ (if the quadratic function intersects the linear function at two points, then point $C$ is the one with the smaller $y$-coordinate).