How to extract six boundary corner points (vertices) of hexagonal region plot?

Question

I need to extract the six boundary corner points (vertices of hexagon) from the following hexagonal region plot.

RegionPlot[
 Abs[(5 x)/Sqrt[3] + y] <= (4 \[Pi])/3 && 
  Abs[x/(3 Sqrt[3]) + y] <= (4 \[Pi])/9 && 
  Abs[-((2 x)/Sqrt[3]) + y] <= (2 \[Pi])/3, {x, -2, 2}, {y, -2, 2}]

Can anyone kindly help me in this regard? Thanks in advance.

Answer 1

Clear["Global`*"];
reg = ImplicitRegion[
   Abs[(5 x)/Sqrt[3] + y] <= (4 \[Pi])/3 && 
    Abs[x/(3 Sqrt[3]) + y] <= (4 \[Pi])/9 && 
    Abs[-((2 x)/Sqrt[3]) + y] <= (2 \[Pi])/3, {x, y}];
bdr = BoundaryDiscretizeRegion[reg];
pts = MeshCoordinates[bdr];
ctr = RegionCentroid[bdr];
corners = {#, EuclideanDistance[ctr, #]} & /@ pts // 
   SortBy[Last] // #[[-6 ;;]] &

Show[bdr, Graphics[{Red, Point@corners[[All, 1]]
   , Blue, Point@ctr
   }
  ], Frame -> True
 ]