19. (12 points) Figure 1 is a planar figure composed of rectangle $A D E B$ , right triangle $A B C$ , and rhombus $B F G C$ , where $A B = 1 , B E = B F = 2$ , $\angle F B C = 60 ^ { \circ }$ . Fold it along $A B$ and $B C$ so that $B E$ and $B F$ coincide, and connect $D G$ , as shown in Figure 2. (1) Prove: In Figure 2, points $A , C , G , D$ are coplanar, and plane $A B C \perp$ plane $B C G E$ . (2) Therefore, from the known condition we have $( x - 2 ) ^ { 2 } + ( y - 1 ) ^ { 2 } + ( z - a ) ^ { 2 } \geq \frac { ( 2 + a ) ^ { 2 } } { 3 }$ , equality holds if and only if $x = \frac { 4 - a } { 3 } , y = \frac { 1 - a } { 3 } , z = \frac { 2 a - 2 } { 3 }$ . Thus the minimum value of $( x - 2 ) ^ { 2 } + ( y - 1 ) ^ { 2 } + ( z - a ) ^ { 2 }$ is $\frac { ( 2 + a ) ^ { 2 } } { 3 }$ . From the given condition we have $\frac { ( 2 + a ) ^ { 2 } } { 3 } \geq \frac { 1 } { 3 }$ , solving gives $a \leq - 3$ or $a \geq - 1$ .
Solution:
19. (12 points)\\
Figure 1 is a planar figure composed of rectangle $A D E B$ , right triangle $A B C$ , and rhombus $B F G C$ , where $A B = 1 , B E = B F = 2$ , $\angle F B C = 60 ^ { \circ }$ . Fold it along $A B$ and $B C$ so that $B E$ and $B F$ coincide, and connect $D G$ , as shown in Figure 2.\\
(1) Prove: In Figure 2, points $A , C , G , D$ are coplanar, and plane $A B C \perp$ plane $B C G E$ .\\
(2)
Therefore, from the known condition we have $( x - 2 ) ^ { 2 } + ( y - 1 ) ^ { 2 } + ( z - a ) ^ { 2 } \geq \frac { ( 2 + a ) ^ { 2 } } { 3 }$ ,\\
equality holds if and only if $x = \frac { 4 - a } { 3 } , y = \frac { 1 - a } { 3 } , z = \frac { 2 a - 2 } { 3 }$ .\\
Thus the minimum value of $( x - 2 ) ^ { 2 } + ( y - 1 ) ^ { 2 } + ( z - a ) ^ { 2 }$ is $\frac { ( 2 + a ) ^ { 2 } } { 3 }$ .
From the given condition we have $\frac { ( 2 + a ) ^ { 2 } } { 3 } \geq \frac { 1 } { 3 }$ , solving gives $a \leq - 3$ or $a \geq - 1$ .