How do you divide an angle into thirds? How a corner is divided into thirds

The trisection of an angle is one of the three major geometric drawing problems proposed by the ancient Greeks 2,400 years ago, that is, the use of a compass and a ruler to divide an arbitrary angle into threes. The difficulty lies in the limitations of the tools used in the drawing. The ancient Greeks demanded that geometric drawings should be made only with straightedges (rulers without scales, only straight lines) and compasses.

This is a question that has attracted many people to study, but none of them have succeeded. In 1837 Van Zier (1814-1848) used algebraic methods to prove that this is an impossible problem of ruler drawing.

In the process of studying the trisectal angles, special curves such as mussel lines, heart lines, and conic curves were discovered. It was also found that as long as the precept of ruler drawing was abandoned, the trichotomy was not a difficult problem. The ancient Greek mathematician Archimedes (287 BC, 212 BC) found that the problem could be solved by fixing a ruler a little.

The method is as follows: add a little p to the edge of the ruler, and the end of the ruler is 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; Make point O move in the CA extension**, point P move in the circumference, and when the ruler passes through B, connect the OPB (see figure).

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.

There is another method of mechanical drawing that can be divided into three equal angles, which is briefly described as follows:

As shown in the figure on the right: ABCD is a square, let AB move evenly to CD in parallel, AD turns to DC in a clockwise direction with D as the center, if AB arrives at DC when DA also happens to arrive at DC, then the trajectory of their intersection point AO is called a third.

Let A be any point on the AC arc, we want to divide the ADC in thirds, let DA and the three-point line AO intersect R, pass R as the parallel line of AB to cross AD, BC in A, B, let T and U be the third equal points of AD, and cross T and U as the parallel lines of AB to cross the three-point line AO in V and W, then DV and DW must be ADC in three equal parts.

Freehand drawing, the protractor is almost fine.

Or do you use a ruler to make a drawing?

That's an unresolved problem.

Three major problems in ruler and gauge drawing, three equal angles, turning a circle into a square, and doubling the cube!

The latest method is the segmented angular method, which can arbitrarily divide any angle. The key point is to the m power where the longitudinal height is set to 2.

The latest method is the segmented angular method, which can arbitrarily divide any angle. The key point is to the m power where the longitudinal height is set to 2.

Trisected angles were one of the three major geometric problems of ancient Greece. The trisect angle is a famous problem in the ancient Greek geometric ruler diagram, and the problem of square and double cube is one of the three major problems of ancient mathematics, and now it has been proved 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.

The latest method is the segmented angular method, which can arbitrarily divide any angle. The key point is to the m power where the longitudinal height is set to 2.