Locating the intersections of two circles in a GeometricScene where the points are symbolic

Question

I'm trying to intersect two circles in a GeometricScene.

segmentMidPoint=.;segmentMidPoint=Function[{a,b},
Block[{steps,l,c1,c2,i1},
    steps={};
    l=Line@{a,b};
    c1=Circle[a,EuclideanDistance[a,b]];
    AppendTo[steps,GeometricScene[{a,b},{l,c1}]];
    
    c2=Circle[b,EuclideanDistance[a,b]];
    AppendTo[steps,GeometricScene[{a,b},{l,c1,c2}]];
    
    i1=First@RegionIntersection[c1,c2];
    Print@RegionIntersection[c1,c2]
]];
Print@segmentMidPoint[a, b];

Up to the second GeometricScene, the scene is interpreted correctly.

Then, the intersection seems not to be interpreted.

I was able to create the intersection with specified points:

Block[{a, b, l, c1, c2, ml},
 a = {1, 0};
 b = {4, 0};
 l = Line@{a, b};
 c1 = Circle[a, EuclideanDistance[a, b]];
 c2 = Circle[b, EuclideanDistance[a, b]];
 Graphics@Line@RegionIntersection[c1, c2][[1]]
 ]

But I wish to do it with symbolic points (a, b, etc.).

Answer 1

Your code uses GeometricScene in ways that I have never considered and which I find confusing, so I'm not sure I understand your question, but why not something like this?

scene =
   GeometricScene[{{a, b, c}, {r}},
     {l == Line[{a, b}],
      c == Midpoint[{a, b}],
      r == EuclideanDistance[a, b],
      Circle[a, r],
      Circle[b, r]}];

examples = GeometricScene[RandomInstance[scene, 3, RandomSeeding -> 1]]

You can extract the values that were given to the points a, b, c with

DeleteCases[examples["Points"], Rule[C["GeometricPoint"][_] ,_], All]

{{a -> {0.432794, -2.27932}, b -> {1.0131, -0.0627347}, c -> {0.722947, -1.17103}}, 
 {a -> {0.284467, -5.31396}, b -> {-6.17663, 0.0374258}, c -> {-2.94608, -2.63827}}, 
 {a -> {3.65394, -2.12234}, b -> {-1.41588, -0.547701}, c -> {1.11903, -1.33502}}}

Update

The following is added to address concerns raised by the OP in a comment to this answer.

How about this? It not only makes the construction, it makes the inference that m is the midpoint. Also, note how I tell GeometricScene to locate intersections.

scene =
  GeometricScene[{{a, b, p, q, m}, {r}},
   {l1 == Line[{a, b}],
    r == EuclideanDistance[a, b],
    c1 == Circle[a, r],
    c2 == Circle[b, r],
    p ∈ c1, p ∈ c2,
    q ∈ c1, q ∈ c2,
    GeometricAssertion[{p, q}, "Distinct"],
    l2 == Line[{p, q}],
    GeometricAssertion[{l1, l2}, {"Concurrent", m}]}];

example = RandomInstance[scene, RandomSeeding -> 1]

FindGeometricConjectures[example]["Conclusions"]

{m == Inactive[Midpoint][{q, p}], 
 m == Inactive[Midpoint][{a, b}], 
 GeometricAssertion[{Line[{a, m, b}], Line[{q, m, p}]}, "Perpendicular"], 
 Inactive[PlanarAngle][{a, m, p}] == 
   Inactive[PlanarAngle][{a, m, q}] == 
   Inactive[PlanarAngle][{b, m, p}] == 
   Inactive[PlanarAngle][{b, m, q}] == 90 °}