Ternary Diagram: Plotting Lines, Areas, Conditions

Question

I am trying to replicate the diagram below with grid lines, lines, areas, and labels (corners, coordinates, lines, and areas).

The blue line can be characterized by a line between the points {0.5,0,0.5} to {0.5,0.5,0} or x=.5. The red line can be characterized by a line between the points {0.5,0.5,0} to {0,0.5,0.5} or y=.5. The black line can be characterized by a line between the points {0.5,0,0.5} to {0,0.5,0.5} or x=.5.

Point "a" is just {1/3,1/3,1/3}.

The orange area is y>=0.5 with x+y+z=1. The grey area overlaps partially with z+0.5y>=0.5 and x+y+z=1.

I identified two possibilities (triangle, TernaryListPlot) but got stuck with each.

Option 1: Triangle

Here the coordinates are off and labeling/grid lines seem intensive work:

Show[Graphics[{Line[{{0, 0}, {1, 0}, {.5, Sqrt[3]/2}, {0, 0}}],RGBColor[0, 1, 0], Line[{{0.5, 0}, {0, 0.5}}]}]]

Option 2: TernaryListPlot

Here the diagram structure is there but everything is based on data points and I can't transition from points to lines and areas.

TernaryListPlot[{{0.15, 0.30, 0.55}, {0.25, 0.70, 0.5}, {0.40,0.50,0.10}, {0.80,0.0,0.20}, {0.55, 0.35, 0.10}, {0.30, 0.45, 0.25}, {0.15, 0.40, 0.45}, {0.80, 0.15, 0.05}, {0.45,0.30,0.25}, {0.25, 0.10, 65}}, FrameLabel -> {"A", "B", "C"}]

Thank you in advance for any hints and help!

Answer 1

a = {1/3, 1/3, 1/3};
b = {0.2, 0.2, 0.6};
c = {0.4, 0.2, 0.4};
TernaryListPlot[{
  {a -> "a"}
  , {Callout[b, "b", Below, Automatic, Appearance -> "CurvedLeader"]}
  , {Callout[c, "c", Above, Appearance -> "Balloon"]}}
 , PlotStyle -> {Black, None, None}
 , FrameTicksStyle -> 
  Directive[FontFamily -> "Times", FontSize -> 14]
 , GridLines -> True, GridLinesStyle -> LightGray
 , FrameTicks -> Range[0, 1, .2]
 , Prolog -> {{Orange, 
    Polygon[{{0, 0, 1}, {0.5, 0, .5}, {0, 0.5, 0.5}}]}
   , {Directive[{Opacity[0.8], GrayLevel[.75]}], 
    Polygon[{{0, 0, 1}, {1, 0, 0}, {0.5, 0.5, 0}}]}
   , {Red, Thick, Line[{{0, 0.5, 0.5}, {0.5, 0, 0.5}}]}
   , {Black, Thick, Line[{{0.5, 0, 0.5}, {0.5, 0.5, 0}}]}
   , {Blue, Thick, Line[{{0, 0.5, 0.5}, {0.5, 0.5, 0}}]}
   , {Text[Style["A", 15], {0, 0, 1}, {6, 2}]}
   , {Text[Style["B", 15], {1, 0, 0}, {-6, 2}]}
   , {Text[Style["C", 15], {0, 1, 0}, {0, -4}]}}, 
 PlotRangePadding -> 0.2]

Appendix (reply to comment)

By use the similary way in my another answer (https://mathematica.stackexchange.com/a/272686/72111),we can plot the region in TernaryListPlot.

Clera[ternary,reg,reg1,sol];
ternary[{p1_, p2_, p3_}] = {p1 + 1/2 p2, Sqrt[3]/2 p2};
reg = ImplicitRegion[{4 x >= .5, z + 5 y >= .5, {x, y, z} >= 0, 
    x + y + z == 1}, {x, y, z}];
reg1 = ParametricRegion[{{x + y/2, Sqrt[3]/2 y}, {4 x >= .5, 
     5 y + (1 - x - y) >= .5, x >= 0, y >= 0, 1 - x - y >= 0}}, {x, 
    y}];
sol = {x, y, z} /. 
   FindInstance[{x, y, z} ∈ reg, {x, y, z}, 10];
TernaryListPlot[sol, 
 Prolog -> {LightBlue, 
   DiscretizeRegion[TransformedRegion[reg, ternary]]}, 
 PlotStyle -> Red]
TernaryListPlot[sol, Prolog -> {LightBlue, DiscretizeRegion[reg1]}, 
 PlotStyle -> Red]