[alogo] Minimal inscribed hexagon

Construct the minimal in perimeter/area equilateral hexagon, inscribed in a triangle ABC and having two vertices on each side.

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From a point Q on AB draw parallel QP to BC. Take QR, PV equal to PQ. Construct the rhombi QPSR, SPVT and RSTU. This defines the equilateral hexagon QPVTUR. Draw lines AU, AT and their intersections H, G with BC respectively. Define the homothety with center A and ratio k = HG/UT. DEFGHI is the homothetical of QPVTUR under this homothety.

The minimal hexagon is symmetric and its center of symmetry coincides with the triangle-center X(37). Look at Hexadivision.html for a more general construction of inscribed equilateral hexagons.

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