How to deduce circle theorems in Wolfram language?

Question

I can deduce the first one by applying FindGeometricConjectures on the following scene:

GeometricScene[{
    "A","B","C"
},{
    Line[{{"A","B"},{"B","C"}}],CircleThrough[{"A","B","C"},Midpoint[{"A","C"}]]
}]

Next one:

GeometricScene[{
    "A","B","C","D"
},{
    Line[{{"A","B"},{"B","C"},{"A","D"},{"D","C"}}],CircleThrough[{"A","B","C","D"}]
}]

But the following is not working:

GeometricScene[{
    "A","B","C","D"
},{
    Line[{{"A","B"},{"B","C"},{"A","D"},{"D","C"}}],CircleThrough[{"A","B","C"},"D"]
}]

Answer 1

It seems that MMA can not find the general result but a special situation such as PlanarAngle equal to 30 Degree.

Here is my test.

FindGeometricConjectures[
 RandomInstance@
  GeometricScene[{"A", "B", "C", 
    "D"}, {GeometricAssertion[{"C", "D"}, {"SameSide", 
      Line[{"A", "B"}]}], CircleThrough[{"A", "B", "C"}, "D"], 
    PlanarAngle[{"A", "C", "B"}, "Counterclockwise"] == 30 \[Degree], 
    Line[{{"D", "A"}, {"D", "B"}}]}]]