Question

A square is inscribes in an equilateral triangle . Find the ratio of the square to that of the triangle?

Answer 1

AD is angular bisector of A. Hence angle GAC or DAC = 30 deg. If we take the point at which square touches AC as K, then angle AKG = 60 deg, since angle AGK = 90 deg.
$tanAKG=\frac{AG}{GK}$
$GK=\frac{a}{2}$, since AD also divides the side of square into half.
$therefore,tanAKG=\frac{AG}{\frac{a}{2}}=\sqrt{3}$ because AKG = 60 deg.
$\to AG=\sqrt{3}\times \frac{a}{2}$