Answer by eldo for Finding maximum or minimum of implicit functions

Also just for fun (in case you don't like to solve equations):

cp = ContourPlot[x^2 + y^2 == (2 x^2 + 2 y^2 - x)^2, {x, -1, 2}, {y, -1, 1},   AspectRatio -> Automatic];x = Cases[cp, {_Real, _Real}, Infinity];points = Point /@ {Take[SortBy[x, First], 2], First@SortBy[x, Last],    Last@SortBy[x, First], Last@SortBy[x, Last]};Show[cp, Graphics[{PointSize@0.025, points}]]