Five basic drawing methods: making a line segment equal to a known line segment, making an angle equal to a known angle, making a middle vertical line of a known line segment, making an angle bisector of a known angle, and making a known straight line pass through a point. The following are the basic methods available for ruler drawing, also known as drawing method. Any step of ruler drawing can be decomposed into the following five methods: making a straight line by knowing two points. Known center and radius can be used as a circle. If two known straight lines intersect, the intersection point can be found. If a known straight line intersects a known circle, the intersection point can be found. If two known circles intersect, the intersection can be found. The problem that rulers and rulers can't draw is the problem that they can't draw with rulers and rulers. Among them, the most famous is the classic problem known as the three difficult problems of geometry: ■ Angle trisection problem: bisecting any angle; ■ Cube problem: Make a cube, so that its volume is twice that of the known cube; ■ The problem of turning a circle into a square: make a square so that its area is equal to the area of a known circle. The above three problems were put forward in ancient Greece 2400 years ago, but they could not be solved under the restriction of Euclidean geometry. It was not until 1837 that the French mathematician Mancer proved for the first time that "angle bisection" and "bicubic" were not problems in drawing ruler. Then in 1882, German mathematician Lin Deman proved that π is a transcendental number, and "turning a circle into a square" was also proved to be a problem that a ruler could not draw. There are two other famous problems: regular polygon method. Use only rulers and compasses to make regular pentagons. Use only rulers and compasses to make a regular hexagon. Making a regular heptagon with a ruler and compasses is a seemingly simple problem, but many famous mathematicians are at a loss because a regular heptagon can't be made with a ruler. Using only rulers and compasses to make a correct nonagon is impossible, because using only rulers and compasses is not enough to divide an angle into three equal parts. Thinking of solving problems: Gauss got the regular drawing method of regular heptagon in his sophomore year, and gave the conditions for drawing regular polygons: the number of sides of the regular drawing polygon must be the product of the non-negative integer power of 2 and different Fermat prime numbers, which solved the unsolved problem for two thousand years. ■ Only compasses are allowed to divide the circumference of a known center into four parts. This question was refuted by Napoléon Bonaparte, which is a challenge to all French mathematicians. Drawing with rusty compasses (that is, compasses with fixed radius) ■ Just use a ruler and rusty compasses to make regular pentagons ■ Drawing with rusty compasses, two points A and B are known, and find a point C to make AB = BC = CA. ■ Given two points A and B, just use a compass with a fixed radius to find C, so that C is the midpoint of line AB. ■ Ruler drawing is based on the ancient Greek idea of "as simple as possible". Can you express it more concisely? Follow this line of thought, and you will have a more concise expression. /kloc-in the 20th century, a mathematician proposed to draw with a ruler and a compass with a fixed radius. 1672 proves that if "doing a straight line" is interpreted as "doing two points in a straight line", then what a ruler can do can be done only with a compass! Several cases of making a new point from a known point: two arcs intersect, a straight line intersects with an arc, and two straight lines intersect. If there is already a circle, then whatever the ruler can do, you can do it with the ruler alone! . A nonagon can be drawn with a ruler.