Regarding the tripartite angle, who solved the question of the tripartite angle

It is completed in four steps according to the Liang's three-point angle operation.

In the 4th century BC, Ptolemy I made Alexandria his capital. He relied on his superior geographical environment to develop maritime ** and handicrafts, and rewarded academics. He built the large-scale "Palace of the Arts" as a center for academic research and teaching; He also built the famous Library of Alexandria, with a collection of 750,000 volumes.

Ptolemy I was well aware of the importance of developing a scientific culture, and he invited famous scholars to Alexandria, where many famous Greek mathematicians were present.

On the outskirts of Alexandria there was a circular villa where a princess lived. There is a river in the middle of the round villa, and the princess's living room is built right in the center of the circle. A gate was opened on the north and south walls of the villa, and a bridge was built over the river, and the location of the bridge was exactly in a straight line with the location of the north and south gates.

The goods that the king gave every day were brought in from the north gate, first in the warehouse at the south gate, and then the princess sent someone to retrieve the apartment from the south gate.

One day the princess asked her attendant, "Which is the longer way from the north gate to my bedroom, or from the north gate to the bridge?" The attendant didn't know, so he hurried to measure it, and the result was that the two roads were as far away.

After a few years, the princess's sister, the little princess, grew up, and the king wanted to build a villa for her. The little princess proposed that her villa should be built like her sister's villa, with a river, a bridge, and a north and south gate. The king was full of promises, and the construction of the little princess's villa soon began, but when the south gate was built and the location of the bridge and the north gate was determined, a problem arose

How can you make the distance from the north door to the bedroom as far as the north door to the bridge?

InRuler drawingUnder the premise that there is no solution to this problem.

Septic angles. It was one of the three major geometric problems of ancient Greece. The problem of trisecting arbitrary angles may have appeared earlier than the other two geometric problems, and it is impossible to find relevant records in history.

But there is no doubt that it will appear naturally, and we ourselves can imagine it now. It has been proved that there is no solution to this problem when judging height under the premise of drawing a ruler and gauge.

Definition. In order to illustrate the sufficient and necessary conditions for the possibility of drawing a ruler and gauge.

The first thing you need to do is to translate geometric problems into an algebraic language. The premise of a plane drawing problem is always given some plane figures, for example, points, lines, angles, circles, etc., but a straight line is determined by two points, an angle can be determined by its vertices and a point on each side for a total of three points, and a circle is determined by a point at the center and circumference of the circle.

Problem 1: How to divide an angle into three with a ruler Dividing an angle into three equal parts is a famous problem in the ancient Greek geometric ruler diagram, and the problem of square and double cube is listed as one of the three major problems in ancient mathematics, and now mathematics has confirmed that this problem is unsolvable. The full description of the issue is:

A given angle is divided into three equal parts using only a compass and an ungraduated ruler. Under the premise of ruler drawing (ruler drawing refers to drawing with a ruler and compass without scale), there is no solution to this problem. If the conditions are relaxed, such as allowing the use of graduated rulers, or if they can be used in conjunction with other curves, a given angle can be divided into thirds.

Question 2: How do you divide an angle into thirds? **Fig. Add a little p to the edge of the ruler and make the ruler end o.

Let the angle to be trisected be acb, with c as the center of the circle and op as the radius as the intersection of the corners of the semicircle at a,b;

Since OP PC CB, COB AC B 3.

The tools used here are not limited to rulers, and the drawing methods are not in line with the common designation.

Can any angle be divided into thirds? Why?

From a purely mathematical point of view, trisecting arbitrary angles has proven impossible; But from a philosophical point of view, it is possible for any angle to be arbitrarily divided. Because mathematics is only a tool for human beings to describe the world, the entire mathematical system is based on a few basic assumptions. For example, is the digital decimal system inherently existing in the objective world?

Not. Many mathematical problems may be caused by a few basic assumptions that are the foundation of the mathematical edifice, and may be very simple problems in the mathematical system formed by binary, third, and n-base, or such problems do not exist at all.

How to prove that any angle of the third division cannot be drawn with a ruler and a ruler.

Using the counterargument method: given an arbitrary angle a, we first make cos(a), assuming that we can divide a in thirds at this time, then we can make cos(a 3), and according to the formula of the triple angle of the dispersed cos, we can get:

4*cos^3(a/3) -3*cos(a/3) =cos(a)

At this time cos(a 3) =x, then the ternary equation :

4x^3 - 3x - cos(a) =0

If the value of cos(a) is different, the solution of the upper orange twig equation is different.

However, for the vast majority of a, the solution of the equation 4x 3 - 3x - cos(a) =0 will be in the form of [cubic root], i.e. cos(a 3) will be in the form of [cubic root].

However, from an arithmetic point of view, there are only five operations that can be performed on ruler diagrams:

Add, subtract, multiply, divide, open squared.

With these five operations alone, the form of [cubic root] cannot be obtained in any case, so the ruler drawing cannot make the amount of [cubic root];

Therefore, cos(a3) cannot be made;

Therefore, a cannot be divided into three equal parts.

This is the general idea of the proof, if you want to prove it rigorously, you have to write too much, it's not necessary here, after all, it's OK if Yuan Lu Min understands the idea).