Is it Possible to Construct a Triangle with an Angle of 60 Degrees and Sides in a Geometric Progression

It is possible to constrict a Right Angled Triangle With Sides in a Geometric Progression. This triangle can be scaled up to give an infinite number of triangles with sides in a geometric progression.
Is it possible to construct a triangle with a 60 degree angle with sides in a Geometric Progression? If so, then the smallest side must be opposite the smallest angle, and the largest side opposite the largest angle. We obtain the triangle below.

60-degree-triangle-with-sides-in-a-geometic-progression.

Applying the Cosine Rule:
\[a^2 = b^2+c^2-2 \times b \times c \times cos A\]

\[(xy)^2 = x^2+(xy^2)^2-2 \times x \times (xy^2) \times cos 60\]

\[(xy)^2 = x^2+(xy^2)^2-2x(xy^2)cos 60\]

\[x^2y^2 = x^2+x^2y^4-2x(xy^2) \times \frac{1}{2}\]

\[ x^2y^2 = x^2+x^2y^4-x^2y^2\]

\[y^2 = 1+y^4-y^2\]

\[ 0 = y^4-2y^2+1\]

\[0= (y^2-1)^2\]

Hence  
\[y=1\]
.
All the sides are length  
\[x\]
  units and the triangle is isosceles, so no such triangle exists.

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