.. How do I solve this triangular tile puzzle?

When you enter this area, walk to the right towards what looks like a cave, then turn right up the steps. At the top of the stairs, you'll find a triangular puzzle with vine designs on it. But there's a piece missing, and you can't solve it till you get that piece --

-- which you'll find inside the gemstone box in the jewelry display in the museum.

What do I do with the box in the jewelry display?

Now that you have the missing piece, let's solve this thing. The goal here is to move the tiles until the pictures are complete. Read to the end of this hint for a picture showing how it looks when you're done.

Left-clicking on a tile picks it up; right-clicking rotates it.

I usually start with the corners because they have two pieces to match. For example, the tile in the bottom left has a white flower on the left and a thorny leaf thingy on the bottom.

Notice that each plant image consists of two pieces that are slightly different. For example, the thorny leaf thingy has one part with one leaf and one with three. A complete thorny leaf thingy should have one of each sort.

Here's the final design:

Triangular Vines Puzzle

What does this thing do?