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

Construction for Inscribing Equilateral Triangle inside Square with a Coincident Vertex

Let $\Box ABCD$ be a square.

It is required that $\triangle DGH$ be an equilateral triangle inscribed within $\Box ABCD$ such that vertex $D$ of $\triangle DGH$ coincides with vertex $D$ of $\Box ABCD$.

Construction

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.

$\blacksquare$

Sources

• 1986: J.L. Berggren: Episodes in the Mathematics of Medieval Islam
• 1992: David Wells: Curious and Interesting Puzzles ... (previous) ... (next): Abul Wafa ($\text {940}$ – $\text {998}$): $38$