Extr = {x /. #, y /. #} &;
Att =
  Extr[
   NMinimize[{(x - #[[1]])^2 + (y - #[[2]])^2, x - x^3 + y^2 == 0}, {x, y}, AccuracyGoal -> 10, PrecisionGoal -> 8][[2]]] &;

DynamicModule[{loc1 = {0, 0}, pt1 = {0, 0}, pt2 = {1, 0}, loc2 = {1, 0}, pt3= {-1, 0}},
 {
  LocatorPane[Dynamic[{loc1, loc2}],
   ContourPlot[{y^2 - x (x - 1) (x + 1) == 0}, {x, -2, 2}, {y, -2, 2},
     AxesLabel -> Automatic, ImageSize -> 600,
    Epilog -> {Red, PointSize[Large],
      Point[Dynamic[pt1 = Att[loc1]]],
      Green,
      Point[Dynamic[pt2 = Att[loc2]]],
      Yellow, Thick,
      Dynamic[InfiniteLine[{pt1, pt2}]]
    }
    ]
   ],
 Dynamic[{pt1, pt2, pt3}];
  }
 ]

ecp = ContourPlot[y^2 == x (x - 1) (x + 1), {x, -2, 2}, {y, -2, 2}];
ec = RegionNearest[ImplicitRegion[y^2 == x (x - 1) (x + 1), {{x, -2, 2}, {y, -2, 2}}]];

DynamicModule[{pts = {{-1, 0}, {1, 0}, {0, 0}}},
              Panel[Row[{LocatorPane[Dynamic[pts, (pts =
                         Block[{ip = ec /@ Most[#], sol},
                               sol = {[FormalX], [FormalY]} /.
                               NSolve[{[FormalY]^2 == [FormalX]
                                                       ([FormalX] - 1) ([FormalX] + 1),
                                       [FormalY] ==
                                       InterpolatingPolynomial[ip, [FormalX]]},
                                      {[FormalX], [FormalY]}];
                               Append[ip, First[Pick[sol, Normalize[Chop[Min /@
                                      DistanceMatrix[sol, ip], 1.*^-6], Max], 1.]]]];) &],
               Show[ecp,
                    Graphics[{{Yellow, Thick, Dynamic[InfiniteLine[Most[pts]]]},
                              {PointSize[Large],
                               {Red, Dynamic[Point[pts[[1]]]]},
                               {Green, Dynamic[Point[pts[[2]]]]}},
                              {PointSize[Medium], Brown, Dynamic[Point[pts[[3]]]]}}],
                    ImageSize -> Medium], Appearance -> None],
               Pane[Dynamic[Grid[Transpose[{{Style["Point 1:", Red, Large],
                                             Style["Point 2:", Green, Large],
                                             Style["Point 3:", Brown, Large]},
                                            Style[#, Large] & /@ pts}]]]]}]]]

ecr = ImplicitRegion[y^2 == x (x - 1) (x + 1), {{x, -2, 2}, {y, -2, 2}}];
BlockRandom[SeedRandom["elliptic"]; (* for reproducibility *)
            (* Quiet suppresses a few harmless error messages *)
            {p1, p2} = Quiet[RandomPoint[ecr, 2]];]

p3 = Chop[EllipticExp[EllipticLog[p1, {0, -1}] + EllipticLog[p2, {0, -1}], {0, -1}]];

ContourPlot[y^2 == x (x - 1) (x + 1), {x, -2, 2}, {y, -2, 2},
            Epilog -> {{Orange, {Thick, InfiniteLine[{p1, p2}]},
                                {Dashed, Line[{{1, -1} p3, p3}]}},
                       {PointSize[Large], {Red, Point[p1]}, {Green, Point[p2]}},
                       {PointSize[Medium], Brown, Point[p3]}}]

Chop[Det[PadRight[{p1, p2, {1, -1} p3}, {3, 3}, 1]]]
   0

ecp = ContourPlot[y^2 == x (x - 1) (x + 1), {x, -2, 2}, {y, -2, 2}];
ec = RegionNearest[ImplicitRegion[y^2 == x (x - 1) (x + 1), {{x, -2, 2}, {y, -2, 2}}]];

DynamicModule[{pts = {{-1, 0}, {1, 0}, {0, 0}}},
              Panel[Row[{LocatorPane[Dynamic[pts, (pts =
                         Block[{ip = ec /@ Most[#], sol},
                               sol = {[FormalX], [FormalY]} /.
                               NSolve[{[FormalY]^2 == [FormalX]
                                                       ([FormalX] - 1) ([FormalX] + 1),
                                       [FormalY] ==
                                       InterpolatingPolynomial[ip, [FormalX]]},
                                      {[FormalX], [FormalY]}];
                               Append[ip, First[Complement[sol, ip, SameTest->(Norm[#1-#2]<1*^-5&)]]]];) &],
               Show[ecp,
                    Graphics[{{Yellow, Thick, Dynamic[InfiniteLine[Most[pts]]]},
                              {PointSize[Large],
                               {Red, Dynamic[Point[pts[[1]]]]},
                               {Green, Dynamic[Point[pts[[2]]]]}},
                              {PointSize[Medium], Brown, Dynamic[Point[pts[[3]]]]}}],
                    ImageSize -> Medium], Appearance -> None],
               Pane[Dynamic[Grid[Transpose[{{Style["Point 1:", Red, Large],
                                             Style["Point 2:", Green, Large],
                                             Style["Point 3:", Brown, Large]},
                                            Style[#, Large] & /@ pts}]]]]}]]]