[Math] Cutting a hexagon to make an equilateral triangle

combinatoricsgeometrypuzzle

The problem is to cut a regular hexagon into parts that can be put together (without overlaps or wasting any parts) to make an equilateral triangle. The cuts should all be straight.

What is the smallest number of parts that will still let you achieve this?

Best Answer

Solution discovered by Harry Lindgren (1961)

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My explanation on how to compute all points :

  1. $A,B,C$ are the vertices of an equilateral triangle
  2. $M$ is the middle of $AC$ and $N$ is the middle of $BC$
  3. $i$ is the projection of $M$ on $AB$
  4. $P$ is the point between $i$ and $B$ such that $|iP|=|iM|$
  5. $Q$ is the point between $A$ and $P$ such that $|QP|=|AM|$
  6. Consider the triangle $ABN$. Let $\delta$ the bisector line of the angle $A$ ($NAB$ to be precise). $R$ is intersection of $PM$ and the parallel to $\delta$ through $Q$.
  7. $S$ is the point between $R$ and $M$ such that $|RS|=|QR|$
  8. $T$ is the point between $R$ and $S$ such that $|RT|=2|SM|$
  9. $V$ is the point such that $TV=QS$ (QSVT is a parallelogram)
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