Points of symmetry of tessellation. I was given this irregular hexagon:  
Then I was told to tessellate it:
Now, I am being asked to find all the points on the hexagon (first picture) which are points of symmetry of my tessellation (second picture).
I don't completely understand what it means by "points of symmetry". I tried researching but couldn't find any relevant results.
Help would be appreciated.
Thank you :)
 A: I'm somewhat surprised the actual question here doesn't seem to be answered yet, the question being "What exactly is a point of symmetry?"
So, what is it? When we have drawings on the plane with some repeating structure (Escher drawings, tesselations, etc.), we tend to say that they're symmetric, but what exactly do we mean by this? Usually, it is meant that these drawings, or structures, contain some reflections or rotations around some point that 'preserve' the structure. We will only focus on rotations.
Since this is rather vague, let's start with a simple example: a plane filled with equilateral triangles whose sides correspond. (Please draw some part of this plane for yourself; if I'd been any better at creating pictures, I'd have done so.) This creates a tesselation with lots of possible rotations. Let's just pick one of many triangles and look at one of its vertices. If we were to rotate the entire plane around this point over $60^\circ$, the entire structure would be preserved - all lines and points in the tesselation would land on lines and points of the tesselation, respectively. The same holds for $120^\circ,180^\circ\ldots,300^\circ$. Since there's some rotation over an angle different from $0^\circ$ that preserves the tesselation structure, we say that this is a 'point of symmetry' for this tesselation.
So, in general, a point of a tesselation is called a point of symmetry if there's a rotation ($\neq 0^\circ$) around this point that preserves the structure. Just to make sure this is clear, consider the following questions:
1) Choose an edge of one of these equilateral triangles. Can you see why rotating over $180^\circ$ around this point preserves the structure? (Hence making this point a point of symmetry.)
2) Is the center of the equilateral triangle a point of symmetry?
*3) Are there any other points of symmetry?
To get to the specific example given in your case, the answer has actually been given in the comments. Since your reaction indicates that you didn't quite see why yet, let's consider one of these in more detail. First of all, we assume only one of the hexagons is given and we'll give the midpoint of $CD$ a name: $P$. If we rotate the hexagon over $180^\circ$ around $P$, what happens? The point $C$ becomes the point $D$ and vice versa. The line segment $DE$ will be rotated around $P$ and will now start at the original point $C$. (This will be a lot clearer, if you look at the image you posted in your question.) Moreover, angles don't change when rotating, so the angle $\angle CDE$ will still be the same. Only, it will now be at the point that was originally called $C$ and lie on the 'other side' of the line segment $CD$. We can do the same for $CB$ and from there for the entire original hexagon. Can you see why the result has to be one of the other hexagons you drew? And why the rest of the structure will thus be at the same place where it originally was as well?
