## Is it Possible to Construct a Triangle with an Angle of 60 Degrees and 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.

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.