Paragraph: A circle C of radius 1 is inscribed in an equilateral P...

Slope of line joining centre of circle to point $D$ $$=\frac{\frac{3}{2}-1}{\frac{3 \sqrt{3}}{2}-\sqrt{3}}=\frac{1}{\sqrt{3}}$$ [makes an angle $30^{\circ}$ with $X$-axis] $\therefore$ Points $E$ and $F$ will make angle $150^{\circ}$ and $-90^{\circ}$ with $X$-axis.
$\therefore E$ and $F$ are given by and $\therefore$ \begin{aligned} \frac{x-\sqrt{3}}{\cos 150^{\circ}} & =\frac{y-1}{\sin 150^{\circ}}=1 \\ \frac{x-\sqrt{3}}{\cos \left(-90^{\circ}\right)} & =\frac{y-1}{\sin \left(-90^{\circ}\right)}=1 \\ E & =\left(\frac{\sqrt{3}}{2}, \frac{3}{2}\right) \\ and F & =(\sqrt{3}, 0) \end{aligned}