Intercept theorem
The intercept theorem (also known as the first, second and third set of beams ) or Four track record is one of the most important statements of elementary geometry. It deals with track conditions and makes it possible for many geometrical considerations to calculate unknown path lengths.
In the synthetic geometry of the first two sets of rays can be generalized with restrictions mutatis mutandis to affine translation planes and apply fully to desarguesche levels. In contrast, the third set of beams, which is called in synthetic geometry, three-jet rate applies, in general, only for pappussche levels, → see affine translation plane # Strahlensatz and dilations.
Formulation of radiation rates
When two by a point ( vertex ) running at right two parallel lines are cut that does not go through the vertex, then the following statements hold:
The first set of beams thus refers to the ratios of beam sections, the second to the conditions of radiation and parallel sections and the third on the ratios of parallel sections.
Comment ( reversal of the radiation law ):
The name set of beams is explained by the fact that one often considered only the special case in which the two parallels are on the same side of the crown ( "V- figure" ). Because then you need to formulate any two intersecting at an apex straight, but only two rays with common origin.
Related geometric concepts
The radiation rate is closely related to the concept of geometric similarity. The triangles ZAB and ZA'B ' are two sketches similar to each other. This means in particular that corresponding aspect ratios in these triangles match - a statement that it follows directly from the set of beams.
Another concept that is related to the theorem is that of the central extension ( of a specific geometric figure). In the left-hand sketch for example, the central dilation forms with center Z and elongation factor (Figure Factor) 1.5 points A and B on the points A 'and B' from. The same applies to the right sketch; here the stretching factor is equal to -0.5.
A similarly close relationship to vector calculations. The calculation rule
For two vectors and a real scalar is just another expression for the radiation rate, because it is then:
Here denotes the length ( Euclidean norm ) of the vector
Simple Example
A simple example of the application of the radiation law is to go back to the ancient Greek philosopher and mathematician Thales of Miletus. This was determined by means of a rod by measuring the shadow length, the height of the Egyptian pyramid of Cheops. In other languages the set of beams is therefore often referred to as a theorem of Thales.
The following example calculation determines the height of the Great Pyramid using the radiation law, but it does not correspond probably the exact calculation of the valley itself:
- Height of the bar:
- Shadow length of the rod:
- Directly measurable length of the shadow of the pyramid:
- Side length of the pyramid:
- Total length of the shadow of the pyramid:
- Seeking height of the pyramid:
Evidence
Set 1
The solders of D and B on the line have the same length, as to be parallel. These solders are heights of the triangles CDA and CBA, which have the associated basic page together. Therefore applies to the areas and continue or area united.
Thus then the following applies:
Applying the standard formula for calculating the size of triangles () returns then
Shorten supplies and.
Solving both after and sets the right sides equal results
Or converted for the track conditions, each on a beam:
Sentence 1 - proof by Archimedes
In addition to the above evidence, based on a representation of Euclid's Elements ( Book 6, L.2 engl. ) Goes back even shorter and more elegant proofs were possible in Ancient Greece. It is enough to show the equality of a case of the possible proportions. The other result from this immediately. Euclid himself only proves a case.
Here's the proof will not be quoted, but executed only in accordance with the Archimedean methodology:
With the conventional lateral and angular designations for the triangles ABZ A'B'Z and shown in the drawing the top ( for the formulation of the radiation blocks) it is shown that a: a '= b: b' is considered. The angles and ' and and ' are the same as the step angle. Denote the heights, which are given by the perpendicular from Z to the line, with h and h ' and their foot points H and H'. Since equal to ' have ' distant ' cathetus and hypotenuse in the two right triangles AHZ and A'H'Z the same relationship to each other. ( In ' modern ' formulation: the same as opposite side of your hypotenuse )
So h: b = h ': b' and therefore h: h ' = b: b'.
From the same ' follows by appropriate consideration of the triangles HBZ and H'B'Z the equation h: a = h': a ' or h: h' = a: a '. And finally, a: a '= b: b'. What was to be proved.
Set of 2
Construct an additional parallel to by A. This parallel cuts in G. Thus applies by construction and Theorem 1 applies to the rays through B also
Converse of Theorem 1
Adopted and would not be parallel. Then there is a parallel that passes through the point and cuts the beam in (*). Since by assumption, it follows
On the other hand, applies after the first set of beams also
This means that and both lying on the beam and have the same distance. This, however, the two points are the same, ie. This is a contradiction to the fact that it should be according to condition ( *) involve two different points. So the assumption that the non-parallelism leads to a contradiction and therefore can not be right; or in other words: It must be.
Applications and generalizations
- Thumb jump, estimating the distance to the theorem by means of his own thumb
- Jacob's staff and foresters triangle are simple measuring instruments according to the principle of the radiation law.
- In the ray optics the rays sentences describe the magnification ratios in a pinhole camera and - together with the lens equation - with an error-free thin lens.
- The statements of the first and second beam can be set to specific nichtdesarguesche levels, the affine translation planes, generalized in synthetic geometry.