One cut – 2 equal parts. How? 🤔

You have a rectangle with a rectangular hole. Can you divide the whole shape into two equal parts with just one straight cut? It might seem tricky, but it’s possible!
In this video, we use an interesting property of a rectangle to show exactly how it’s done.

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[ineedprimos]

Circle: yes. Hexagon: yes. Triangle: no (I'm not confident with this one)

[victor_ah]

this will always work with polygons with an even number of sides, but it does not work in all cases with polygons with an odd number of sides

[anuj_person_]

It would not work with the triangle I think.

[thecowilsoninc]

There's a rule. There is always 1 straight line that can cut 2 2D objects equally in half. This applies to 3D objects too. There is always 1 plane that can cut 3 3D objects in half. This goes on.

[Flairex]

If it were an equlateral triangle, the line would have to pass through both the center and one of the vertices. The hexagon would work because it has symmetry like the rectangle, and the circle is symmetrical on every side, so it would work as well

[_rd_]

You can actually cut any N objects equally into halves with a single cut in an N-dimensional space

Works for 2 any objects in 2D, 3 things in 3D and goes on for any theoretical higher dimension

[PARASF]

The center points of both hole and rectangle must be coordinated at same point. So any line crossing over center can divide the rectangle except triangle.

[sir.op]

It works on any shape with a hole of any shape, you just need to find their centers of mass (2 points define a line)

[valerium]

It always works regardless of the shape of the hole, number of holes, shape of the shape itself, number of shapes... Not only that, but there is no point on the shape for which there is not a line that bisects it, passing through the point.

Proof: Take an arbitrary shape, and an arbitrary point. Draw a line through the point such that it splits the shape in two regions. These regions have areas we will call a1 and a2.

Now, if you rotate the line about the point, the areas a1 and a2 will change continuously. If you rotate the line 180 degrees, then area a1 will be equal to what was area a2. Since these areas changed continuously, then that means that if it was the case that a1 > a2, and now a2 > a1, then there is a point where a1 = a2, and vice versa.

[johnallegood]

It would with the circle and regular hexagon, but for the triangle it would only work on special cases. With an equilateral triangle, it would also have to pass through either a vertex or the center of a side.

[zushyart]

If it was an isosceles triangle pointed towards or away from the rectangle’s center, there would be a valid straight cut. Otherwise, the triangle wouldn’t be split in half along its altitude.

[gestroyer]

It allways works regardless of the holes shape since you can just cut besides the hole so that the areas are equal. In case of the rectangle in the video you could cut straight below the rectangle. The cuts shown are very elegant solutions but not the only ones.

Anyways nice vid ❤

[parthchatupale]

Every line passing through the centre of a triangle do not divide it in the two equal areas, same goes for any polygon having odd no of corners as @victor_ah said.And hence above method won't work for odd cornered polygons.

[epigone]

If a polygon has an even number of vertices with equal opposing angles or sides, then any line that passes through a center of that polygon will divide it in two equal polygons (because each "half" will have an equal amount of vertices with matching angles and sides).

[tecnicoyt]

The circle works (of course),

I think the hexagon works as well,

The triangle *can* work, the most straight forward example is when the triangle is upright below the middle point of the bigger rectangle, making a vertical cut a perfect one, but I'm not sure if it would always work (I think not (...?))

[SmartGamer]

hexagon & circle
(assuming regular hexagons)

also works with any even-sided polygon!
(assuming it’s regular)

[zsombororovec]

When the haxagon is right, it will work only than or if it is not right but well placed to be true this applies for the triangle too, meanwhile the rectengal works because the shape and so as the circle.

[gaiet]

It would qork with hexagon and circle the same way, but with triangle the triangle would need to be:
1 - symetric
2 - cut on the symmetry line

[DEGRODANTONTHEINTERNET-ufbr]

Sandwich cut or something like that theorem. As long as there are 2 shapes of any and all shape, size, position or orientation, there will always be 1 line that will divide both into equall pieces. This theorem also works with 3 shapes in 3d, or theoretically 4 shapes in 4d.

[MarcelFuentesAlonso-dsjb]

Circle:yes (same reason as the rectangle)
Hectagon:yes (same reason as the rectangle)
Triangle: depends a lot on the form and placement, if the triangle has 3 equal sides he must be in the center or in a place where the diagonals pass trough iy center, if the traingle has only 2 equal sides he must be in the center only (i can't calculate well the simetry of a 2 equal sides traingle please correct me if i'm wrong) and if the traingle doesn't has any equal side it doesn't matter where you place it, it doesn't has simetry so its imposible