Equilateral Triangle Problem #2

$\text{From the figure, }\angle ADM = \angle AEM = 90\degree \Rightarrow \angle ADM + \angle AEM = 180\degree.\newline\Rightarrow AEMD \text{ is a cyclic quadrilateral. }\hspace{20pt}\text{[ Sum of opposite angles of quadrilateral is }180\degree.\text{]} \quad\dots \textbf{1}$

$\text{Also, }\angle ADM = \angle AFM = 90\degree \Rightarrow \text{ angle subtended by }AM \text{ at point }D\text{ and }F \text{ are equal. }\newline\Rightarrow AMDF \text{ is a cyclic quadrilateral. }\quad\dots \textbf{2}$

$\text{Using } \textbf{1}\text{ and }\textbf{2,}\,\text{ we get } AEMDF \text{ is a cyclic pentagon with one of the diameter of circumscribing circle }AM.\newline\text{So, }G\text{ is the centre of circumscribing cirlce.}\newline\Rightarrow GE = GF = GD = \text{ radius } \quad\dots\textbf{Eq. 1}$

$ED\text{ is a chord of circumscribing circle }\Rightarrow \angle EFD = \angle EAD = 30\degree\hspace{20pt}\text{[ Angle subtended by chord in the same segment are equal ]}\newline\text{Similarly, }FD\text{ is a chord of circle }\Rightarrow\angle FED = \angle FAD = 30\degree$

$\text{Now, in }\triangle EFD,\,\angle EFD = \angle FED = 30\degree \newline\hspace{65pt}\Rightarrow DE = DF \hspace{10pt} \text{[ Sides opposite to equal angles are equal ]}\quad\dots\textbf{Eq. 2}$

$\text{In quadrilateral }AEDF\text{, } \angle EDF = 180\degree - \angle EAF = 180\degree - 60\degree = 120\degree.\hspace{12pt}\dots\textbf{Eq. 3}$

$\text{Using }\textbf{Eq. 1},\,\textbf{Eq. 2}\text{ and }GD\text{ being the common side in }\triangle GED \text{ and }\triangle GED,\,\triangle GED\cong\triangle GFD. \newline\Rightarrow \angle GDE = \angle GDF = 60\degree\hspace{12pt}\dots\textbf{Eq. 4}$

$\text{Using }\textbf{Eq. 1}\text{ and }\textbf{Eq. 4},\newline \angle GED = \angle GFD = 60\degree\hspace{12pt}\dots\textbf{Eq. 5}$

$\text{Using }\textbf{Eq. 4}\text{ and }\textbf{Eq. 5}\text{ we get, }\triangle GED\text{ and }\triangle GDF \text{ are equilateral triangles. }\newline\text{So, in quadrilateral }GEDF\text{ all sides are equal. Hence it is a rhombus.}$