Theorem of Thalès

In certain countries of Europe, whose France, the theorem of Thalès indicates a Théorème Géométrie which, in dimension 2, affirms that a straight line parallel with the one on the sides of a triangle divides this last in a similar triangle (see stated precise below). In other languages, in particular in English, this result is known under the name of theorem of intersection .

This result is allotted to the Mathématicien and Greek Philosophe Thalès de Milet. This attribution is explained by a legend according to which Thalès would have calculated the height of a pyramid by measuring the length of the shade on the ground of this one and that of the shade of a stick height given. However, of the historical documents prove that the similarities of the triangles had already noticed by the Babylonian . But the first found written demonstration of this theorem is given in the Éléments of Euclide (proposal 2 of book VI). The latter rests on the proportionality of surfaces of triangles equal height (see below the detail of the proof).

The theorem of Thalès spreads in higher dimension. The result is equivalent to results of projective geometry like the conservation of the birapport by projections or the property of Pappus. On a more elementary level, the theorem of Thalès is used to calculate lengths in trigonometry, on the condition of having two parallel straight lines. This property is used in instruments of calculation lengths.

In English, the theorem of Thalès indicates another theorem of geometry which affirms that a triangle registered in a circle and whose east coast a diameter is a right-angled triangle.

Statements and teaching

In practice, the theorem of Thalès makes it possible to calculate reports/ratios length and to highlight relations of proportionality in the presence of parallelism. He is thus taught in France with the college:

Theorem of Thalès: Is a triangle $ABC$, and two points D and E of the right-hand sides $(AB)$ and $(AC)$ so that the line $(OF)$ is parallel to the right-hand side $(BC)$ (as indicated on the figure below). Then one a:

$\backslash \; frac\; \{AD\}\; \{AB\}\; =\; \backslash \; frac\; \{AE\}\; \{AC\}\; =\; \backslash \; frac\; \{OF\}\; \{BC\}$.

Two possible configurations of the theorem of Thalès.

The theorem of Thalès shows that triangles ABC and ADE are homothetic: there exists a Homothétie of center has sending B on D and C on E . One of the reports/ratios given above is with the sign close the report/ratio of homothety. More precisely, the report/ratio of homothety is $+AD/AB$ in the first configuration and $-AD/AB$ in the second. The theorem of Thalès is sometimes stated more simply while affirming than a straight line parallel with one on the sides of the triangle intersects this triangle in a similar triangle.

It can be implemented in various geometrical construction industries utilizing compass and rule. For example, it can justify a construction making it possible to divide a segment into a given number of equal shares.

To be more rigorous, the statement given above requires the use of a Euclidean Distance to give a direction to the lengths mentioned ( AB , BC ,…). A more exact statement can be (and will be) given within the framework of the geometry closely connected. Within this framework, the concept length is replaced by that of algebraic Mesure, and only the report/ratio has a direction (see further).

Reciprocal theorem

The theorem of Thalès (in dimension 2), in its direct direction, makes it possible to deduct certain proportions as soon as a certain parallelism is known. Its Réciproque makes it possible to deduce a parallelism as soon as the equality of certain reports/ratios is known.

Reciprocal of the theorem of Thalès: In a triangle ABC, let us suppose given points $D$ and $E$ belonging respectively to the segment $$ and $$. If reports/ratios AD/AB and AE/AC are equal, then the lines (DE) and (BC) are parallel.

It is to be noticed that the demonstration of this reciprocal results from the theorem. Indeed, let us consider a E' point of the segment $$ such as (DE') that is to say parallel to (BC). Then the points has, E', C are aligned in this order and AE'/AC = AD/AB = AE/AC thus it comes that AE' = AE. However there does not exist that only one point located between has and C checking this property thus E' = E. Consequently, (OF) = (DE') is parallel to (BC).

Theorem of the mediums

See also: Theorem of the mediums

The theorem of the mediums is a specialization of the theorem of Thalès, for which the points D and E correspond to the mediums of the segments $$ and $$. If a line passes by the mediums on two sides of a triangle, it is parallel to the right-hand side which supports the third side; and the length uniting the mediums of the two east coasts equalizes with half the length on the third side:

Theorem of the right-hand side of the mediums: Is a triangle ABC , and name D and E the respective mediums of $$ and $$. Then the lines $(OF)$ and $(BC)$ are parallel and one a: $2.DE=BC$

The reciprocal one of the theorem of Thalès justifies that the two lines are parallel; moreover, the theorem of Thalès applies and it comes:

$\backslash \; frac\; \{OF\}\; \{BC\}\; =\; \backslash \; frac\; \{AD\}\; \{AB\}\; =\; \backslash \; frac\; \{1\}\; \{2\}$.

Teaching and names

This theorem is known under the name of theorem of Thalès in the Enseignement of mathematics in France. More precisely the result in the first configuration and the theorem of the right-hand side of the mediums are taught as of the Classe of fourth Frenchwoman and the “theorem of Thalès” strictly speaking and its reciprocal in the Classe of third Frenchwoman. Geometrical constructions implementing the theorem must be seen with the college. The bond between the theorem of Thalès and homotheties must be only taught with the Lycée. In his report/ratio, critical Jean-Pierre Kahane openly the absence of the congruence of triangles in the teaching of mathematics in France, from which it holds like person in charge the reform known as of the modern Mathématiques.

No old text seems to allot the discovery of a result similar to Thalès. The first reference where such an attribution is made is in the Éléments of geometry of Rouche and Comberousse in 1883. One of the causes of this attribution would be the incentive made with the aggregative S at the end of the 19th century century allot to the results of the names of mathematicians so that they are interested in the history of mathematics.

It is as under the name Théorème from Thalès as the result is known in the countries of Méditérannée and the countries of North or Eastern Europe. However, in the countries of English language and in Germany, this result is known under the name of theorem of intersection . Name theorem of Thalès indicates in these countries the property according to which any inscribed angle in a half-circle is right (to read Théorème from Thalès (circle)).

In Switzerland, the theorem is mainly approximate thanks to the “small property of Thalès” such as she is taught in France. The “Swiss theorem of Thalès” on the other hand expresses the square height in a right-angled triangle.

Origins

No proof can attest knowledge or not theorem of Thalès or a similar result before (slow) the appearance of the writing. The first known and undeniable traces of the use of mathematical knowledge are pragmatic texts coming from first great civilizations controlling the writing. The oldest texts treat all Numération, i.e. the art of calculation, particularly the multiplication, division and extraction of roots. It is not astonishing that these texts appear initially: without the control of this art, the theorem of the article does not have utility. The first traces of a knowledge of the theorem or a close substitute go back to III with the Bronze Age at the same time in ancient Egypt and mesopotamy in Babylonian civilization.

Babylonian civilization

See also: mathematical Knowledge in Mésopotamie

During the archaeological excavations at the 19th century and 20th century, the first texts of mathematics (and even first texts written) were found in Mésopotamie. In this area oldest known civilization developed, that of Sumer, which developed the first form of known writing, the sumérien, a wedge-shaped writing consisted of successions of arrows and points. He succeeded the Akkadien S, the Babylonian S and the Assyrian S, which regained this shape of writing, as well as the use of the sexagesimal Système. Babylon created a school of the scribes : thousands of dried shelves of clay, in general written in sumérien, filled with exercises of mathematics were found and studied by the archeologists and historians of mathematics since work of Hilprecht (towards 1855).

The existence of numerical tables and the solution to problem algebraic attest strong knowledge into arithmetic. In comparison, few shelves relate to the Géométrie strictly speaking. But some relate to nevertheless the divisions of right-angled triangles with straight lines. They highlight knowledge on the similarity of the right-angled triangles. However, the problems of geometry are posed by the drawing of an illustrative figure; there does not exist any proof that the concepts of parallelism and right angle were formalized. The shelf of Such Harmal going back to worms -1800 shows a problem of geometry with four overlapping right-angled triangles. The scribe highlights there relations of proportionality between with dimensions ones.

Shelf MLC 1950 (dated between -1900 and -1600) described a exercise in which the scribe seeks to calculate the lengths of the bases of a right-angled trapezoid starting from information on the surface trapezoid, its height and the height of the corresponding triangle. The data are indicated on the figure opposite (in sexagesimal notation). The scribe calculates:

• the half the sum sought lengths as the report/ratio of the surface by the height (of other shelves confirm that the formula giving the surface of the trapezoid was known);
• the half-difference by application of a not explained formula; by noting $A$ the surface of the trapezoid $BDEC$, it is written today literally:

$\backslash \; frac\; \{BC-DE\}\; \{2\}\; =\; \backslash \; frac\; \{has\}\; \{2AD+BD\}$. Roger Caritini explains how to obtain this formula by applying the small property of Thalès to the triangles $ABC$ and $ADE$ on the one hand, and to the triangles $CBA$ and $CFE$ on the other hand. He deduces from this reasoning that the scribe had (thus, the theorem of Thalès or a substitute). However, the problem is presented by its figure without the assumptions being stated;

Possible demonstration of the formula

the parallelogram $BDEF$ admits a right angle in $B$, and two on its sides are by parallel assumptions: it is thus a Rectangle. In particular, the line $(EFF)$ is parallel to $(AB)$. The application of the small property of Thalès to the triangles $CBA$ and $CFE$ gives:

$\backslash \; frac\; \{BC-DE\}\; \{OF\}\; =\; \backslash \; frac\; \{data\; base\}\; \{AD\}$ or $\backslash \; frac\; \{BC-DE\}\; \{data\; base\}\; =\; \backslash \; frac\; \{OF\}\; \{AD\}$.

Like the line $(OF)$ is parallel to the right-hand side $(BC)$, a new da application the property of Thalès gives:

$\backslash \; frac\; \{AD\}\; \{AB\}\; =\; \backslash \; frac\; \{OF\}\; \{BC\}$ or $\backslash \; frac\; \{BC\}\; \{AB\}\; =\; \backslash \; frac\; \{OF\}\; \{AD\}$.

From where one can deduce:

$BC+DE=\; (AB+AD).\; \backslash \; frac\; \{BC-DE\}\; \{data\; base\}\; =\; \backslash \; frac\; \{2.A\}\; \{data\; base\}$.

Consequently, one from of deduced:

$\backslash \; frac\; \{BC-DE\}\; \{2\}\; =\; \backslash \; frac\; \{has\}\; \{2AD+BD\}$.

Ancient Egypt

See also: Mathematical in ancient Egypt

Egyptian civilization, only four papyruses offer mathematical solutions to problem. The Egyptologists indirectly deduce mathematical knowledge from ancient Egypt of the administrative documents treating from believed of the Nile, of the calculation of the taxes, the cultivable ground distributions, the drawing of the fields after the destruction of the reference marks following the risings,… According to some, the Pyramides of Gizeh would show a geometrical knowledge put for the use of architecture. However, according to others, the absence of numerical tables shows a weak interest for mathematics in addition to the aspects applicatifs.

Most famous of the four papyruses is the Papyrus Rhind named according to Alexander Henry Rhind (1833 - 1863) , a Scottish antique dealer which buys it in 1858. He would have been written by the scribe Ahmès under the Pharaon Apophis Aouserré (towards -1550) taking again the contents of a papyrus not found writes under the reign of the Pharaon Amenemhat II (towards -1850).

Sylvia Couchoud, a Egyptologist studies this papyrus with attention. In addition to information which it provides on knowledge into arithmetic and algebra, it offers the statement of the theorem of Thalès, called seqet , applied to a numerical example.

Ancient Greece

Ancient Greek civilization is different from that of Egypt or Babylon. Philosophy and the beauty are essential subjects there. It is thus not astonishing that Greek mathematics does not have any more, for primary goal, the solution to problem pragmatic but theoretical. Pythagore establishes a Géométrie founded on principles , which will become later Axiome S to reach an experimental but purely speculative and intellectual approach not . For Plato, mathematics constitutes the base of the teaching of the King-philosopher of the ideal city. The geometry takes really its roots in Greek civilization.

A vision of this nature radically modifies the formulation of the theorem of Thalès, which one finds the first demonstration written known in the Eléments of Euclide. Three essential components changed. The theorem is stated in a perfectly general way, unlike the Egyptians who describe this result using an example, or Babylonians who seem to use the result implicitly. It is shown, previously the treaties of mathematics were presented in the form of a succession of capable techniques to find the good performance. The concept of demonstration was absent. Finally the reciprocal one is stated, which is also a first.

The calculation height of a pyramid, a legend

Literary texts of Greek Antiquity refer to work of Thalès de Milet to the VI E, whose no writing reached us. However, no old text allots the discovery theorem at Thalès. In its comment of the Elements of Euclide, Proclos affirms that Thalès would have brought back the result of its voyage in Egypt. Hérodote brings back the same thing and precise which it is one of the seven wise founders of this civilization. A famous anecdote reports that Thalès obtained the admiration of Pharaon by measuring the height of one of the pyramids. Plutarque indicates that:

This legend is taken again by other authors. Diogène Laërce written:

Bernard Vitrac puts serious doubts about the reality of the precise details: . Plutarque speaks just about proportionality between the heights of the pyramid and the stick on the one hand, the lengths of their projected shades on the other hand. Laërce evokes the equality on the adjacent sides to a right angle of an isosceles right-angled triangle. The legend according to which Thalès would have invented " son" theorem while wanting to calculate the height of a pyramid today is conveyed and embroidered by many Internet sites, many newspapers of popularizations, and by certain authors.

According to Michel Greenhouses, this history was used and transmitted in ancient Greek civilization like a Moyen mnemotechnics to remember result: .

This history attests also probable origin of the geometry: the calculation of the distances and the sizes characteristic of inaccessible objects (here the height of a pyramid) probably led the man to wonder about the relations of the distances between benchmarks. This interrogation naturally led it to be interested in trigonometry, from where the emergence of results comparable with the theorem of Thalès. In addition, in the various versions an object of reference is used, an axis, an axle, or Thalès itself, is used like a gnomon , term meaning instrument of the knowledge, comprehension . Some see in the Babylonian origin of this term an additional proof that the statement of the theorem and its demonstration are of Babylonian origin

A version of the embroidered history of discovered result by Thalès could be as follows.

Version of the legend

At the time of a voyage in Egypt, Thalès would have visited the built pyramids several centuries earlier. Admiring these monuments, it would have been put at the challenge to calculate the height of it. Thalès would thus have undertaken a measurement of the pyramids, whose principle would rest on the concept of similar triangles and proportionality. Thalès would have noticed that at that time year, at midday, the shadow of a man or a stick equalized the size of the man or the length of the stick. The sun rays being able to be presumedly parallel, Thalès would have deduced from it that it would be the same for the height for the pyramid and its projected shade.

Encore was necessary it to be able to measure the projected shade: it would have located the top of the projected shade of the pyramid but to measure it in its entirety, it would have been necessary for him to leave the center of the pyramid which was not accessible. Thalès would have profited from an additional advantage: not only the shadow equalized the height of the pyramid but the rays of the sun were perpendicular to an edge of the base. The top of the shade of the pyramid would have been then on the mediator on a side of the base. It would have is enough to him to measure the distance separating the end from the shade and the medium on the side, to add to this length a half-side to obtain the height of the pyramid.

the fact that the top of the shade of the pyramid is on the mediator on a side at midday does not hold absolutely chance but owing to the fact that the pyramids are directed full south or full western. The Pyramide of Khéops is located at a Latitude of 30°, the length of the shade equalizes that of the stick when the sun made 45° with the vertical. The angle which form the sun with the vertical varies during the year between 6,73° (with most extremely of the summer) and 53,27° (with most extremely of the winter) and forms an angle of 45° only twice in the year (on November 21st and on January 20th). It would be an extraordinary chance which Thalès had been there at this precise moment. At any other period of the year, the length of the shade is proportional to the height.

Thalès would have mentioned itself these remarks. It would be turned over and would have explained why the height of the pyramid is proportional to the length of its shade. By comparing the length of the shade and the height of a planted stick, it would have been easy to him to know the proportionality factor and to then apply it to the shade of the pyramid to determine its height of it. Plutarque does not say another thing besides.

Demonstrations

Proof mentioned by Euclide

In the approach of Euclide, the points are indivisible elements from which the geometrical objects are defined. From this point of view, the concepts of segments and lines are not differentiated. The property shown by Euclide is not exactly the theorem as it is quoted nowadays. A translation going back to 1632 is the following one:

Book VI, Proposal 2: “If one meine a straight line parallel with the one of dimensions of a triangle, which cut the two others dimension; it will cut them proportionally; & if both dimension of a triangle are crossed proportionally, the cutting line will be parallel to different dimension. ”
If one carries out a straight line parallel with the one on the sides of a triangle, which cut the two other sides, it will cut them proportionally. And if the two sides of a triangle are cut proportionally, the cutting line will be parallel to the other side.

The data of this theorem are thus:

• a triangle, by definition delimited by three straight lines (segments) AB, BC, and CA;
• a straight line OF parallel to straight line BC intersecting AB in D and AC in E.

The notations are those introduced by Euclide after the statement; the illustration opposite gives the provision of the points. The conclusion given is:

“AD will do with dB what AE is with EC. ”

In other words, in current mathematical writing: $\backslash \; frac\; \{data\; base\}\; \{DA\}\; =\; \backslash \; frac\; \{THIS\}\; \{EA\}$.

The step of Euclide is based on the fact that the surface of a triangle is equal to half the length its height, by-report/ratio at a base (or side) unspecified, multiplied by the length of the base in question. It notes that the heights of the triangles DEB. and DEC by-report/ratio at their common base OF have the same length. These two triangles have consequently the same surface, and a fortiori, they thus have even same the ratio (of surfaces) with any not-null surface, and in particular that of triangle DEA. Like the heights of the triangles DEB. and DEA by-report/ratio, respectively at bases data base and DA, are confused, Euclide from of deduced that the ratio of DEB. by DEA is the same one as the ratio of data base by DA. By analogy, the ratio of DEC by DEA is the same one as the ratio of EC by EA. Proposal 11 of the book V states that the ratios which are the same ones as another ratio are the same ones. Euclide from of thus deduced that the ratio of data base by DA is the same one as the ratio of EC by EA.

The reasoning suggested by Euclide results today in the following equalities:

$\backslash \; frac\; \{data\; base\}\; \{DA\}\; =\; \backslash \; frac\; \{Surface\; (DEB.)\}\{Surface\; (DEA)\}=\; \backslash \; frac\; \{Surface\; (DEC)\}\{Surface\; (DEA)\}=\; \backslash \; frac\; \{THIS\}\; \{EA\}$. The equalities are based on the following observations:

• triangles DEB. and DEA Thus have a common height H resulting from E., their surface is respectively ½ BD×h and ½ DA×h.
• triangles DEB. and DEC have a common base OF, and the opposite tops B and C are by assumptions on a straight line parallel with (OF).
• Lastly, triangles DEC and DEA have a common height h' resulting from D. Donc, their surface is respectively ½ CE×h' and ½ EA×h'.

In the form of Table of proportionality:

Purely vectorial proof

It is necessary to raise the question of the validity of a vectorial demonstration of the theorem of Thalès. Indeed, the vectorial Géométrie is often based on a geometrical definition of the vectors, definition in which the theorem of Thalès plays a paramount role when it is a question of affirming that $k\; (\backslash \; vec\; \{U\}\; +\; \backslash \; vec\; \{v\})\; =\; K\; \backslash \; vec\; \{U\}\; +\; K\; \backslash \; vec\; \{v\}$.

But one can however be interested in a possible writing of the theorem of Thalès and his justification thanks to the vectorial operations. What could make it possible to generalize the theorem of Thalès to all Espace Euclidean refines associated with a vector Space.

To say that D is on (AB) is to write that there exists a reality X such as $\backslash \; overrightarrow\; \{AD\}\; =x\; \backslash \; overrightarrow\; \{AB\}$.

In the same way, to say that E is on (AC), it is to write that there exists a reality there such as $\backslash \; overrightarrow\; \{AE\}\; =y\; \backslash \; overrightarrow\; \{AC\}$.

Lastly, to say that the lines (ED) and (BC) are parallel, it is to write that there exists real T such as $\backslash \; overrightarrow\; \{OF\}\; =\; T\; \backslash \; overrightarrow\; \{BC\}$.

The preceding equalities and the Relation of Chasles make it possible to write that:

$y\; \backslash \; overrightarrow\; \{AC\}\; =\; \backslash \; overrightarrow\; \{AE\}$

$y\; \backslash \; overrightarrow\; \{AB\}\; +\; there\; \backslash \; overrightarrow\; \{BC\}\; =\; \backslash \; overrightarrow\; \{AD\}\; +\; \backslash \; overrightarrow\; \{OF\}$

$y\; \backslash \; overrightarrow\; \{AB\}\; +\; there\; \backslash \; overrightarrow\; \{BC\}\; =x\; \backslash \; overrightarrow\; \{AB\}\; +\; T\; \backslash \; overrightarrow\; \{BC\}$

The writing according to the vectors $\backslash \; overrightarrow\; \{AB\}$ and $\backslash \; overrightarrow\; \{BC\}$ must be single because these vectors are not colinéaires. Thus $y=x\; \backslash ,$ and $y=t\; \backslash ,$

The three equalities are thus obtained:

$\backslash \; overrightarrow\; \{AD\}\; =x\; \backslash \; overrightarrow\; \{AB\}$

$\backslash \; overrightarrow\; \{AE\}\; =x\; \backslash \; overrightarrow\; \{AC\}$

$\backslash \; overrightarrow\; \{OF\}\; =x\; \backslash \; overrightarrow\; \{BC\}$.

The advantage of this statement and this demonstration is that does not oblige to treat the various cases of configuration mentioned above.

Generalizations of the theorem of Thalès

Parallel cases of three straight lines

Always in dimension 2, the theorem of Thalès can be stated in an equivalent way:

Theorem of Thalès: Is two lines $d$ and $d\text{'}$ (Euclidean real plan) and three straight lines parallel $(AA\text{'})$, $(BB\text{'})$ and $(CC\text{'})$ intersecting $d$ and $d\text{'}$ respectively in $A$ and $A\text{'}$, $B$ and $B\text{'}$, and $C$ and $C\text{'}$, as indicated on the figure opposite. Then:

$\backslash \; frac\; \{AB\}\; \{AC\}\; =\; \backslash \; frac\; \{A\text{'}\; B\text{'}\}\; \{A\text{'}\; It\}$.

This conclusion is equivalent to the one of the two following equalities: $\backslash \; frac\; \{AB\}\; \{BC\}\; =\; \backslash \; frac\; \{A\text{'}\; B\text{'}\}\; \{B\text{'}\; It\}$ or $\backslash \; frac\; \{BC\}\; \{AC\}\; =\; \backslash \; frac\; \{B\text{'}\; It\}\; \{A\text{'}\; It\}$; or with: $\backslash \; frac\; \{A\text{'}\; B\text{'}\}\; \{AB\}\; =\; \backslash \; frac\; \{B\text{'}\; It\}\; \{BC\}\; =\; \backslash \; frac\; \{A\text{'}\; It\}\; \{AC\}$.

The first statement given of the theorem of Thalès is the specialization of the second if two points are confused (in general $A$ and $A\text{'}$). By considering the parallel with $d\text{'}$ passing by the second statement has results from the first.

Dimensions higher than 2

Often stated like a theorem of plane geometry, the theorem of Thalès spreads without difficulty in higher Dimension, in particular in dimension 3. The use of parallel straight lines is replaced by parallel hyperplanes; the lines (d) and (of) do not have to be presumedly coplanar:

Theorem of Thalès : Is $(d)$ and $(of)$ two lines of same a Espace refines; and $H\_a$, $H\_b$ and $H\_c$ three Hyperplane S parallel. It is supposed moreover that the line (d) respectively intersects the hyperplanes in $A$, $B$, $C$, and just as the line (d') respectively in $A\text{'}$, $B\text{'}$ and $C\text{'}$. Then, one a:

However, the proof mentioned by Euclide, which uses explicitly the concept of surface, is specific to dimension 2 and does not spread with higher dimension. In the same way, there is not the reciprocal obvious one with this general statement. In a note, where reciprocal is mentioned giving the existence of parallel hyperplanes.

Proof using a projection refines

In its book, Marcel Berger proposes an algebraic proof of the version of the theorem of Thalès in higher dimension. Its proof is not based on an axiomatic approach of the geometry, but on the current definition of Espace refines and vectorial.

The proof of Marcel Berger consists in building a bijection refines F between the two lines D and of sending $A$ on $A\text{'}$, $B$ on $B\text{'}$, and $C$ on $C\text{'}$. It is possible to deduce from the definition of an application closely connected that a bijection refines between right-hand sides closely connected preserves the report/ratio. Thus:

To build F indeed, Marcel Berger proposes to provide the unit with the hyperplanes parallel with $H\_a$ of a structure line closely connected (structure quotient), so that the application which with a point of D (or of of ) associates the hyperplane passing by this point and parallel with $H\_a$ that is to say an application closely connected. Without introducing the concept of space quotient, one can also define the application F like the restriction on D of projection on the line $d\text{'}$ parallel to $H\_a$. By definition, F sends indeed $A$ on $A\text{'}$, $B$ on $B\text{'}$, and $C$ on $C\text{'}$. The reverse of F is defined as the restriction on the right-hand side $d\text{'}$ of projection on $d$ parallel to $H\_a$.

Conservation of the Birapport S by projections

The Birapport is a projective invariant associated with four points. The theorem of conservation of the birapports by projection is dependant of close with the theorem of Thalès, one being able without much evil to result from the other.

Just as the three parallel straight lines of the theorem of Thalès can be replaced by parallel hyperplanes in a space closely connected of size higher than 2, the four convergent lines of this theorem of conservation of the birapports can be replaced by hyperplanes belonging to same a beam in higher dimension.

The interest from this point of view is to underline the analogy of the “report/ratio” $\{\backslash \; overline\; \{AB\}\}\; \backslash \; over\; \{\backslash \; overline\; \{AC\}\}$ intervening in the theorem of Thalès with the birapport used in projective geometry: first invariant by a line transformation closely connected is left closely connected towards another exactly like second is left invariant by a projective projective line transformation towards another.

The theorem of Thalès in the plans arguésiens

It is also possible to give a demonstration of the theorem of Thalès from axiomatic of the plans closely connected ones released to the 20th century; if one takes the party to present the plans closely connected (on a commutative body) like the plans arguésiens checking the property of Pappus, the theorem of Thalès can be proven.

Applications of the theorem of Thalès

Results of projective geometry and relationship with homotheties

In Geometry, the theorem of Thalès or its reciprocal can be used to establish conditions of alignment or parallelism. Without calling upon the concepts projective line, they make it possible to obtain satisfactory versions of the results actually raising of the projective Géométrie. The theorem of Thalès can be used as substitute of the Homothétie S in the demonstrations.

• Theorem of Ménélaüs: being given a triangle ABC and three points A', B' and It belonging respectively to right-hand sides (BC), (AC) and (AB); the points A', B' and It are aligned if:

$\backslash \; frac\; \{\backslash \; overline\; \{A\text{'}\; C\}\}\; \{\backslash \; overline\; \{A\text{'}\; B\}\}.\; \backslash \; frac\; \{\backslash \; overline\; \{B\text{'}\; has\}\}\; \{\backslash \; overline\; \{B\text{'}\; C\}\}.\; \backslash \; frac\; \{\backslash \; overline\; \{It\; B\}\}\; \{\backslash \; overline\; \{It\; has\}\}\; =1$

• Théorème of Ceva: being given a triangle ABC and three points A', B' and It belonging respectively to right-hand sides (BC), (AC) and (AB); lines (AA'), (BB') and (CC') are concourrantes or parallel if:

$\backslash \; frac\; \{\backslash \; overline\; \{A\text{'}\; C\}\}\; \{\backslash \; overline\; \{A\text{'}\; B\}\}.\; \backslash \; frac\; \{\backslash \; overline\; \{B\text{'}\; has\}\}\; \{\backslash \; overline\; \{B\text{'}\; C\}\}.\; \backslash \; frac\; \{\backslash \; overline\; \{It\; B\}\}\; \{\backslash \; overline\; \{It\; has\}\}\; =-1$

• Théorème of Pappus: Is two lines D and of; three points has, B and C of D; three points A', B', and This of. One notes P, Q, and R the respective intersections of (AB') and (A' B), of (B' C) and (BC'), and of (AC') and (A' C). Then the points P, Q and R are aligned.
• Theorem of Desargues: Is two triangles ABC and A' B' It such as lines (AB) and (A' B') is parallel, the same for (BC) and (B' It) and for (AC) and (A' It). Then lines (AA'), (BB') and (CC') are parallel or convergent.

Constructible numbers

See also: constructible Number

A question which was raised during Antiquity, and in particular in the form of the problem of the Quadrature of the circle, is the possibility of building a figure using the rule (not graduated) and of the compass:

• the rule is a idealized instrument making it possible to consider a line passing by two already traced points;
• the compass is a idealized instrument making it possible to consider a circle of center a point already built and ray the carryforward of a distance carried out between two already built points.

A point of the Euclidean plan is known as constructible with the rule and the compass if it can be obtained by a finished number of stages starting from the points of coordinates (0,0) and (0,1). Following work of Georg Cantor, it can be marked that all the constructible points with the rule and the compass are of number countable.

A constructible number is a real number which can be obtained like constructible punctual coordinate. The whole of the constructible numbers is stable by nap, product and opposite. The theorem of Thalès shows that the product of two constructible numbers is a constructible number. Indeed, for two real nonnull constructible $x$ and $y$, a calculation gives the justification of construction opposite:

$\backslash \; frac\; \{xy\}\; \{X\}\; =\; \backslash \; frac\; \{there\}\; \{1\}$

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