This reply might be a bit late. But why make it harder than it is

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From your drawing

$\displaystyle \angle{DAE} = 180 - 15 - 15 = 150$

$\displaystyle \angle{ADB}=\angle{AEC} = 90 - 15 = 75$

$\displaystyle \angle{EAC}=\angle{DAB} = \beta$

If $\displaystyle \bigtriangleup{ABC}$ is equilateral set all its angles to $\displaystyle \alpha$

This gives us:

$\displaystyle 150 + 2*\beta + \alpha = 360 \Rightarrow \alpha = 210 - 2*\beta$

$\displaystyle 75 + \beta + (90-\alpha) = 180 \Rightarrow \beta = 15 - \alpha$

$\displaystyle \alpha = \frac{210-30}{3} = 60$

Which proves that $\displaystyle \bigtriangleup{ABC}$ is equilateral.