Poncelet-Steiner's theorem

from Wikipedia, the free encyclopedia

The set of Poncelet and Steiner is a set of the synthetic geometry . It is based on a conjecture by Jean Victor Poncelet from 1822 and was proven by Jakob Steiner in 1833 .

The sentence says that every construction that can be carried out with a compass and ruler can only be carried out with a ruler, provided that a solid circle and its center are given.

Steiner proves the theorem by showing how the elementary tasks of construction can be carried out using only those limited aids.

The sentence has a counterpart in Mohr-Mascheroni’s sentence , which states that constructions that can be carried out with a compass and ruler - i.e. the construction tasks considered in classical ancient geometry according to Euclid - can also be carried out with the compass alone.

Constructions

Draw a parallel line to a circle diameter through any point

Construction according to Steiner: draw a parallel to a circle diameter through any point.

Given on a straight line g is the circle diameter AB with its center M and any point P.

  1. First, draw a straight line through points B and P.
  2. Determine point C on the straight line at will.
  3. Draw a straight line through points A and P.
  4. Draw a straight line through points A and C.
  5. Draw a straight line through points M and C, you will get the intersection point D.
  6. Draw a straight line through points B and D, you will get the intersection point E.
  7. Draw a straight line through the points E and P, this results in the parallel to the straight line g.
  • From the adjacent construction it is easy to see (besides the given points A and B there is no further point on the circle) that z. B. for a solution of the problem draw a parallel to a straight line through any point , the circle is not required. According to Steiner, the specification of a straight line with the three points A, M and B, of which point M is in the middle, is sufficient.

Drop a plumb line from any point on the central axis of a circle or erect a vertical line on it

The central axis of a circle with the diameter AB , hereinafter referred to as "central axis AB ", and any point P.

  • Any point P should mean that the point P may also be within the circle, on the circle and on the center axis AB of the circle.
  • Two possible positions of the point P are considered constructively:
a) Point P with sufficient distance to an additional central axis, which should be perpendicular to the first, hereinafter referred to as "second central axis", in order to be able to continue the construction with its help (e.g. JD ); described in variant 1 (variant 1).
b) Point P is too close to a second central axis to be able to continue the construction with its help; described in variant 2 (variant 2).

The construction consists of three building blocks.

  • A parallel to the given line AB .
  • A perpendicular to the parallel just drawn
a) as the second central axis of the circle in Var. 1
b) as a vertical line within the circle in Var. 2.
  • A parallel to the vertical line just drawn from the given point P to the central axis AB or, if the point P lies on the central axis AB of the circle, a parallel through or from the point P.

Variant 1 and variant 2

Variant 1:
Drop a perpendicular from any point onto the central axis of a circle or erect a vertical line on it.
Point P at a sufficient distance from the initially virtual second central axis;
see 4 examples as animations
Variant 2:
Drop a perpendicular from any point onto the central axis of a circle or erect a vertical line on it.
Point P too close to the virtual second central axis;
see 4 examples as animations
  1. First determine point E on the circle by eye with BE ≈ circle radius (angle BME ≈ 60 °). An angle BME of approx. 55 ° to approx. 70 ° is helpful for an easily usable position of the later intersection point J or C in Var. 2.
  2. Draw a straight line from point A through point E.
  3. Draw a straight line from point B through (in var. 2 bis) point E.
  4. Determine the point F on the straight line A through E.
  5. Connect point B with F.
  6. Connect the point M with F, the result is the intersection point G.
  7. Draw a straight line from point A through point G to segment BF , the result is the intersection point H.
  8. Draw a straight line from point H through point E to the circle, this results in the intersection point I or C in Var. 2. The route HI or HC in Var. 2, is, according to Steiner, a parallel to the circle diameter AB .

Continuation of variant 1 (point P with sufficient distance to the initially virtual second central axis of the circle)

  2. Draw a straight line from point A through point I, the result is the intersection point J.
  3. Draw a straight line from point J through point M to the circle, the intersection points C and D. The second central axis JD is now the auxiliary perpendicular for the parallel perpendicular to be determined.

Continuation of variant 2 (point P is too close to the second virtual central axis of the circle to be able to continue the construction clearly)

  2. Draw a straight line from point E through point M to the circle, this will result in intersection point D.
  3. Connect point D with C, you get the intersection M 1 . The line DC is the auxiliary vertical for the parallel vertical to be determined.

Continuation of variant 1 and variant 2

  2. Draw a straight line from point C through point P.
  3. Find point K on the straight line as you wish.
  4. Connect point D with K.
  5. Connect point D with P.
  6. Connect point M or M 1 in var. 2, with K, the point of intersection L results.
  7. Draw a straight line from point C through point L to segment DK , the result is the intersection point N.
  8. Draw a straight line so that points P and N lie on it.

The results that can be achieved with it depend on the position of the given point P.

Results

a) P does not lie on the central axis AB : The plumb line sought has its base point P 'on the central axis AB .
b) P lies on the central axis AB : The desired perpendicular to the central axis AB runs through the point P. In Var.1 it is a parallel to the second central axis JD or in Var.2 a parallel to the auxiliary perpendicular CD .

See also

Individual evidence

  1. ↑ Jakob Steiner: The geometric constructions, carried out by means of the straight line and a solid circle, as a subject of instruction in higher educational institutions and for practical use . Ferdinand Dümmler, Berlin 1833 (accessed April 2, 2013).
  2. ↑ a b Jakob Steiner: The geometrical constructions, carried out by means of the straight line and a solid circle, as a subject of teaching in higher educational institutions and for practical use . Ed .: Ferdinand Dümmler. Berlin 1833 ( ETH Library, ⅠⅠ Constructions using a ruler under certain conditions [page 14 §. 6.], page 15, task Ⅰ. See also panel Ⅰ, Fig.3 [accessed on September 20, 2016]).