Distribution of points within a triangle
using barycentric (area) coordinates

See https://www.johndcook.com/blog/2025/09/11/random-inside-triangle/
and https://leancrew.com/all-this/2025/09/barycentric-coordinates-and-random-distribution-of-points/

3D items

Unit cube
In[]:=
cube={AmbientLight[RGBColor[.75,.75,.75]],Opacity[.25],Cuboid[{0,0,0},{1,1,1}]};
Axes
In[]:=
axis1=Arrow[{{0,0,0},{1.25,0,0}}];​​label1=Text[Style[α,FontSize->14],{1.35,0,0}];​​axis2=Arrow[{{0,0,0},{0,1.25,0}}];​​label2=Text[Style[β,FontSize->14],{0,1.35,0}];​​axis3=Arrow[{{0,0,0},{0,0,1.25}}];​​label3=Text[Style[γ,FontSize->14],{0,0,1.35}];​​
Projection/normalization plane
In[]:=
plane={AmbientLight[RGBColor[.5,.5,.5]],Opacity[.5],Polygon[{{1,0,0},{0,1,0},{0,0,1}}]};
Line from origin to opposite corner
In[]:=
line={Thick,Red,Line[{{0,0,0},{1,1,1}}]};
Intersection of line and plane
In[]:=
point={PointSize[Large],Red,Point[{1/3,1/3,1/3}]};

Cube alone

In[]:=
Graphics3D[{cube,axis1,axis2,axis3,label1,label2,label3},Lighting->None,Boxed->False,ViewPoint->{10,5,4},ViewVertical->{0,0,1},ImageSize->Medium]
Out[]=

Cube and plane

In[]:=
Graphics3D[{cube,axis1,axis2,axis3,label1,label2,label3,plane},Lighting->None,Boxed->False,ViewPoint->{10,5,4},ViewVertical->{0,0,1},ImageSize->Medium]
Out[]=

Intersection of line and plane

In[]:=
Graphics3D[{cube,axis1,axis2,axis3,label1,label2,label3,plane,line,point},Lighting->None,Boxed->False,ViewPoint->{10,-8,4},ViewVertical->{0,0,1},ImageSize->Medium]
Out[]=

Random points projected onto plane

Points
In[]:=
n=10000;​​a=RandomReal[1,n];​​b=RandomReal[1,n];​​c=RandomReal[1,n];
Normalize to the plane
In[]:=
sum=a+b+c;​​x=a/sum;​​y=b/sum;​​z=c/sum;​​pts=Transpose[{x,y,z}];
Plot and view down diagonal
In[]:=
ListPointPlot3D[pts,ViewPoint->{1000,1000,1000},ViewVertical->{0,0,1},BoxRatios->{1,1,1},ImageSize->Medium,AxesOrigin->{0,0,0},Boxed->False,PlotRange->Automatic,Ticks->None,AxesStyle->Directive[Black,AbsoluteThickness[1]],PlotStyle->PointSize[.006]]
Out[]=