public MapPoint StartPoint {get; set;}

// Builder constructors need to run on the MCT.
ArcGIS.Desktop.Framework.Threading.Tasks.QueuedTask.Run(() =>
{
  // retrieve the bezier curve's control points
  using (CubicBezierBuilder cbb = new CubicBezierBuilder(bezierSegment))
  {
    MapPoint startPt = cbb.StartPoint;
    Coordinate2D ctrlPt1 = cbb.ControlPoint1;
    Coordinate2D ctrlPt2 = cbb.ControlPoint2;
    MapPoint endPt = cbb.EndPoint;

    // or use the QueryCoords method
    cbb.QueryCoords(out startPt, out ctrlPt1, out ctrlPt2, out endPt);
  }
});

CubicBezierBuilder cbbEx = new CubicBezierBuilder(bezierSegment);
MapPoint startPtEx = cbbEx.StartPoint;
Coordinate2D ctrlPt1Ex = cbbEx.ControlPoint1;
Coordinate2D ctrlPt2Ex = cbbEx.ControlPoint2;
MapPoint endPtEx = cbbEx.EndPoint;
     
// or use the QueryCoords method
cbbEx.QueryCoords(out startPtEx, out ctrlPt1Ex, out ctrlPt2Ex, out endPtEx);