And the connecting line of the corresponding point is perpendicular to the crease (if DD' is perpendicular to AE, it can be omitted here)

( 1)

This has nothing to do with folding.

AB=6√2

AC= 12

DC=8√3

AE=DC/2=4√3

(2)

This has nothing to do with folding along AE. Draw a sketch yourself and remove that so as not to affect it. ABC doesn't have to draw either

Draw an ACD, fold it along AC, and draw a symmetrical ACD''

It is easy to prove DP = d'' p.

So DP+EP = D'' P+EP.

Because the line segment between two points is the shortest, the formula is the smallest when P is on the straight line between D'' and E.

Note that DD''C' C is actually a regular triangle.

So the minimum value is 12.

(3)

Connect Dubai and DC.

You can find the area of triangle ad' c, and note that AD'B and BD' c are congruent.

So you can find the BD'C area.

The distance from d' to BC is the height of BD'C based on BC.