Inscribing Equilateral Triangle inside Square with a Coincident Vertex/Construction 1
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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 ($\text {940}$ – $\text {998}$): $38$