Masters of translation theory, please enter! Add 100 points if you are good .

It's not a math department, but I guess I understand. The following text:

Our last example is geometric, but from a "decomposition" point of view. For example, a box is not made up of pieces of shape, but is actually made up of many "points". The question is how to recognize the dots (drawing a diagram will help you understand).

Suppose there is a square, divide it with 9 line segments, and each line will divide the square in a ratio of 2:3 area, please prove that at least 3 of the 9 lines will intersect at the same point. Since each line divides the square into 2 quadrilaterals, then each line must intersect the 2 discrete sides of the square, then we must get a trapezoid with at least 2 right angles, and its area is equal to the length of its base edge multiplied by the length of its middle line (trapezoidal area = side length * height).

Obviously, all trapezoids have the same length at the bottom, so their high ratio must also be 2:3But a trapezoid must only have 2 midlines (1 vertically and 1 horizontally), so we get 4 midpoints, 2 on each midline.

Each midpoint becomes the common midpoint of at least 2 trapezoids, and this midpoint must be contained within the known 9 lines. (Let me explain it to you like this: we have 4 midpoints in each trapezoid, and each midpoint has to pass through at least 2 lines, but we know that there are 9 such lines, so there must be 1 midpoint to pass through 3 lines ---the original sentence translation is not so thorough)).

Now that we've proven the above, I'm going to make it a little harder, and I need to twist my brain into a twist to think like a twist. Then there's the pigeon nest principle. (Its simple form is :.)

If n 1 object is placed in n boxes, at least one box contains two or more objects. ）

Our last example is geometry, but here the "division", i.e. the box, does not the geometry of the whole part, in fact, they are in it. The difficulty is to achieve it is that these precise points will help (or have a hunch of those points by doing some drawings).

We have a square and 9 lines running through it. Each wire cutting square.

For the quadrilateral, and the quantitative area of the quadrilateral is 2:3.

Prove that at least three of these 9 lines pass through the same point.

Since each sub-square has 2 quadrilaterals, each line is cut square.

on the edges of non-contiguity. The figure acquires a double right-angled trapezoid, the area of which is equal to its base (here the square side).

Multiply the height of the "middle line" and the line is perpendicular.

Their two edges are parallel and equidistant from the edges.

Because any kind of trapezoidal foundation is the same, than.

The midline must be 3. But there is also a possible midline.

For the trapezoid, so we get four different points and call them the midpoint, 2 for each midline and 2 for the midline

The trapezoids intersect with 9 given lines. Each of these lines must pass through these midpoints. We have 4 points and 9 lines, then there is at least one point to pass through the three lines.

We have shown that given 9 rows, each square cut the area in a trapezoidal shape is in a ratio of 2:3, at least three lines pass through the same point.

Now we get into the more difficult problem, which usually takes a little.

That's how it should be