I don't know the internals, so I am just guessing. The solutions you have in mind are degenerate, as they require two points to coincide. Maybe the random algorithm has zero chance of hitting on them. It keeps finding a degenerate Triangle, so it tries another random instance, and so on.

Thanks, Gianluca, but I need to work with triangles. I don't undertand why Mathematica can find intersection points with circumcircles and perpendicular bisectors or angle bisector but RandomInstance can't find the easy intersection of two lines, (the last one):

Scene1 = GeometricScene[{a, b, c, e}, {Triangle[{a, b, c}],
   AngleBisector[{a, b, c}],
   CircleThrough[{a, b, c}],
   e \[Element] AngleBisector[{a, b, c}],
   e \[Element] CircleThrough[{a, b, c}]
   }]
RandomInstance[Scene1] (*It works well!*)

Scene2 = GeometricScene[{a, b, c, e}, {Triangle[{a, b, c}],
   AngleBisector[{a, b, c}],
   CircleThrough[{a, b, c}],
   PerpendicularBisector[{a, c}],
   e \[Element] PerpendicularBisector[{a, c}],
   e \[Element] CircleThrough[{a, b, c}]
   }]
RandomInstance[Scene2](*It works well!*)

Scene3 = GeometricScene[{a, b, c, e}, {
   Triangle[{a, b, c}],
   e \[Element] AngleBisector[{a, b, c}],
   e \[Element] Line[{a, c}]
   }]
RandomInstance[Scene3](*It works well!*)

Scene4 = GeometricScene[{a, b, c, e}, {
   Triangle[{a, b, c}],
   e \[Element] Line[{a, b}],
   e \[Element] Line[{a, c}]
   }]
RandomInstance[Scene4] (*It fails. The triangle becomes a line!!*)