Part of the construction was done for you. Here are the steps to create this part of the construction.
Start with the objects in the diagram below.

• Draw a circle with radius AB centred at B.

• Mark the points where ⨀A and ⨀B intersect. Call them C and D.

• Since A, C, and D are all on ⨀B, AB=BC=BD. Since B, C, and D are all on ⨀A, AB=AC=AD. This means that AB=AC=AD=BC=BD.

• So, △ABC and △ABD are congruent equilateral triangles.
• Draw the line through A and B.

• Draw the line through A and C.

• Draw the line through A and D.

• Mark the point other than B where ⨀A and AB intersect. Call it E.

• Mark the point other than C where ⨀A and AC intersect. Call it F.

• Mark the point other than D where ⨀A and AD intersect. Call it G.

• Draw the segment between B and C.

• Draw the segment between C and G.

• Draw the segment between G and E.

• Draw the segment between E and F.

• Draw the segment between F and D.

Complete the construction.

• To complete the construction of a regular hexagon inscribed in ⨀A with a vertex at B, carry out the following step:
• Draw the segment between B and D.

This means the six triangles formed by the segments between A and these six points are all isosceles.
Now, since △ABC and △ABD are equilateral triangles, m∠BAC and m∠BAD are both 60°. Find m∠CAG using the fact that these three angles form a straight angle.
m∠CAG = 180°–m∠BAC–m∠BAD
= 180°–60°–60°
= 60°
By the vertical angle theorem, it follows that m∠EAG, m∠EAF, and m∠DAF are all 60°. An isosceles triangle containing a 60° angle is equilateral, so △ABC, △ACG, △AEG, △AEF, △ADF and △ABD are all equilateral. Since each of these triangles has a side in common, they are all congruent equilateral triangles.

• So, all sides and interior angles of hexagon BCGEFD are congruent to each other. In other words, BCGEFD is a regular hexagon.