Connect AH, AG

Then according to the symmetry of equilateral triangle, ah = BH.

So GH+BH = ah+GH

The minimum value of ah+BH is obviously obtained on the line of a, h and g.

That is, the minimum value of GH+BH = the length of line segment AG.

Become AM⊥BC

Obviously, there are cm = BC/2 = 3 and GM = 1.

And am = cm * √ 3 = 3 √ 3.

So according to Pythagorean theorem:

AG=2√7

So the minimum value of GH+BH = segment length AG = 2 √ 7.

For reference! JSWYC