Okay. Proving that you can draw that perfect hexagon is kind of easy since it was easy to draw a regular hexagon.
First, connect the centre of the hexagon with each of the hexagon vertices. Remember this? 
Now do this:
You should have 6 triangles. These triangles are all congruent because of SSS. Each of a hexagon's sides is equal and each triangle shares a side with another, making every side equal in length, We already have three sides.Next, each of these triangles has the same inner angle, since the hexagon splits it evenly, so 360 divided by 6 = 60 degrees. Since these triangles are also isosceles, that means that the other two angles are equal two! That means that each triangle is an equilateral!
You should have 6 triangles. These triangles are all congruent because of SSS. Each of a hexagon's sides is equal and each triangle shares a side with another, making every side equal in length, We already have three sides.Next, each of these triangles has the same inner angle, since the hexagon splits it evenly, so 360 divided by 6 = 60 degrees. Since these triangles are also isosceles, that means that the other two angles are equal two! That means that each triangle is an equilateral!
Now it is pretty obvious. Since the radius of the circle is equal to each of the hexagon's sides, we now know that the hexagon we drew is indeed perfect.
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