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This is a part $3$ of a sequence of questions starting with my highly upvoted question (at the time of writing, my third-best post). Feel free to extend this series using other polygons and fractions.

Let there be an equilateral triangle that has $n+1$ notches on each edge (corners included) to divide each edge into $n$ equal parts. We can make cuts on the triangle from notch to notch. Is it always possible to cut out a connected piece with area $\frac{1}{3}$ the area of the original triangle if $n≥2$? If it is possible for $n$, it is possible for any multiple of $n$.

For simplicity, in my answer, I might use a transformed version of the coordinate system so that the vertices of the triangle are $(0,0),(n,0),$ and $(0,n)$. Since this is a linear transformation, it has constant determinant, so ratios of areas won't be affected.

Since I'm posting a self-answer, I won't show parts of my work here.

mathlander
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1 Answers1

1

$2$: Cut from $(2,0)$ to $(0,1),$ and $(0,2)$ to $(1,0),$ and take the triangle that has one of its sides as $(2,0)$ to $(0,2).$

$3$: Cut from one corner to the other side, then take the smaller piece.

$3n + 1$ for $n \geq 1$: Cut from $(2,0)$ to $(0,1), (0,2)$ to $(1,0),$ and $(0,2)$ to $(2n + 2, n - 1)$

$3n + 2$ for $n \geq 2$: Cut from $(4,0)$ to $(0,2), (0,4)$ to $(2,0),$ and $(0,4)$ to $(2n + 4, n - 2).$

And last but not least, $n = 5$:

Cut from $(0,4)$ to $(2,3), (0,3)$ to $(1,4), (0,1)$ to $(4,1), (0,3)$ to $(3,0),$ and $(3,0)$ to $(3,2).$

mathlander
  • 4,097