Thanks - I found that, and indeed it has some good stuff in there.

Some of it's a bit cryptic, but I'm working through it.

It's a bit rough going.

It can be challenging to keep the syntax, math, geometry, and POV-Ray's
interpretation of a "Vector" all straight.

Still working on the sphere in contact with the 3 points of a triangle and a 4th
point...  Can't seem to get it to work out.


#declare Corner_A_1 = <0, 0, 0>;
#declare Corner_A_2 = <4, 0, 0>;
#declare Corner_A_3 = <4, 4, 0>;
#declare H = 1;


#declare Circumscribed = object {Circumscribe (Corner_A_1, Corner_A_2,
Corner_A_3)};



 #macro Circumscribed_Sphere (Corner_1, Corner_2, Corner_3, Height)  // POINTS
A, B, C
 union{
  #declare Middle =  Triangle_M_out (Corner_1, Corner_2, Corner_3); // POINT P /
Middle
   #declare Radius =  Triangle_R_out (Corner_1, Corner_2, Corner_3); //
   #declare Normal =  Triangle_Normal(Corner_1, Corner_2, Corner_3);

   sphere {Middle, 0.1 translate Normal texture {Plane} }
   object {Vector (Middle, vtransform ( Middle, transform {translate Normal} ),
0.05) texture {Plane}}
   object {Vector (Middle, vtransform ( Middle, transform {translate -4*Normal}
), 0.05) texture {Plane}}

   // translate Middle along Normal by Height to get point D
   #declare PointD  =  vtransform ( Middle, transform {translate Normal*Height}
) ;
   sphere {o, 0.1 translate PointD texture {Object} }   // POINT D
   sphere {Corner_A_3, 0.1  texture {Object} }
   object {Vector (Corner_A_3, PointD, 0.05) texture {Tex_Dark}}


   // midpoint EP
   #declare Midpoint = (Corner_A_3 + PointD)/2;
   sphere {Midpoint, 0.1  texture {Object} }
   #declare Tangent = Triangle_Normal ( Corner_A_3, Middle, PointD );
   // Perpendicular to triangle containing sphere center
   #declare PerpRef = vtransform (Tangent*5, transform {translate Corner_A_3} );
// Just a big tangent
   object {Vector (Corner_A_3, PerpRef, 0.05) texture {Plane}}


   #declare Perp = vcross (PointD-Corner_A_3, Tangent);
   #declare PerpN = vnormalize (Perp);
   #declare PerpV = vtransform (Perp*2, transform {translate Midpoint} );
   object {Vector (Midpoint, PerpV, 0.05) texture {Tex_Dark}}
   #declare Adjacent = vlength (Midpoint - PointD);
   #declare Theta = VAngleD (Midpoint - PointD, Middle - PointD);
   #declare Opposite = Adjacent * atan (Theta);


   #declare Center = vtransform (Midpoint, transform {translate PerpN *
Opposite} );
   sphere {Center, 0.1  texture {Object} }
   #declare SRadius = vlength (Center - PointD);

   sphere {Center, SRadius texture {Shaded} } // Circumscribed sphere
   }
 #end

#declare  Sphere = object {Circumscribed_Sphere (Corner_A_1, Corner_A_2,
Corner_A_3, H)};

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