Inscribing Equilateral Triangle inside Square with a Coincident Vertex/Construction 1

Construction

 * Inscribing-equilateral-triangle-inside-square-1.png

By Construction of Equilateral Triangle, let an equilateral triangle $\triangle ABN$ be constructed on $AB$ such that $N$ is inside $\Box ABCD$.

Let $AB$ be produced to $F$ such that $AB = BF$.

Draw an arc centred at $F$ with radius $FN$ to cut $AB$ at $G$.

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

Then $DGH$ is the required equilateral triangle.

Proof
It is necessary only to note that $N$ passes through $DH$, which is demonstrated in the simpler Construction 4.

Then the proof for that construction can be applied.