Inscribing Equilateral Triangle inside Square with a Coincident Vertex/Construction 2

Construction

 * Inscribing-equilateral-triangle-inside-square-2.png

By Construction of Equilateral Triangle, let an equilateral triangle $\triangle DAI$ be constructed on $AD$ such that $I$ is inside $\Box ABCD$.

Bisect $\angle ADI$ and bisect it again towards $AD$ to cut $AB$ at $G$.

Construct $H$ on $BC$ such that $DH = DG$.

Then $DGH$ is the required equilateral triangle.

Proof
First a lemma:

Lemma
Because $\triangle DAI$ is equilateral, we have that:

The result follows from the lemma.