eqn1  =  
      Eliminate[{(c^2) == (((x + (a/2)))^2) + (y^2), 
          (b^2) == (((x - (a/2)))^2) + (y^2), a + c == 2  b}, {b, 
          c}]

eqn2 = Factor[Part[eqn1, 1] - Part[eqn1, 2]]

a = 1;
<< Graphics`ImplicitPlot`
curva = 
        ImplicitPlot[
          eqn2 == 0, {x, 0, 3}, {y, -2, 
            2},  
DisplayFunction -> 
            Identity];
arco = 
        ImplicitPlot[
          eqn2 == 0, {x, -1/6, 0}, {y, -2, 
            2},  
DisplayFunction -> Identity, 
          PlotStyle -> RGBColor[1, 0, 0]];
lineas  =  
        Line[{{-0.5, 0}, {0, ((Sqrt[3])/2)}, {0.5, 
              0}}];
Show[{Graphics[curva],  Graphics[arco],  
        Graphics[lineas]}, 
AspectRatio -> Automatic, 
      PlotRange -> {{-1, 3}, {-2, 
            2}}, 
DisplayFunction -> $DisplayFunction]

Solve [(eqn2 == 0) /. y -> 0, x]

Solve[eqn2 == 0, y]

g1 = 
        Plot[11 + 60 x - 36 (x^2) - 16 (Sqrt[1 + 6 x]), {x, -(1/6), 
            3}, DisplayFunction -> 
            Identity];
g2 = 
      Plot[(11/36) + ((5 x)/3) - (x^2) + 
          (4/9) (Sqrt[1 + 6 x]), {x, -(1/6), 3}, 
        DisplayFunction -> Identity]
Show[GraphicsArray [{g1, g2}], 
      AspectRatio -> Automatic]