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Beitrag zum Jahr der Mathematik 2008

Neue bisher unbekannte, elementare Formeln für Dreiecke

Wissenschaftliche Studie 2008 174 Seiten

Mathematik - Geometrie






Part I: Triangles
The Theorems of W E H R L E
Area of right-angled triangles A = w + ¼ w*
w = A - r
Product of catheti ab = 2r(2R+r)
Diameter of incircle 2 r = a+b – c
Sum of catheti a+b = 2 (R+r)
Catheti only by radi a= R+r ±√ (R– r – 2rR) = R+r ± √ (R– A)
Theorems of W E H R L E od differences w = (2r) = 4r
The smalles rational triangles with integer sides x= l - m y = l (l - m) and z =m (l +1)
Every rational triangle consists of two rational rectangular ones
The distance of the centers of in- and circum circle d = R(R-2r) or d = √ (R + r - A)
Proving the Theorem of WEHRLE
Trigonomical Wehrles:
The Wehrle-number of the differences
of the sinus-values is (r/R)²
Other Theories of tigomometrical Wehlre
The sum of squares of the sides
( a²+b²+c² ) =2( ab+ac+cb ) –( w* +8w)
Proof for the area-theorem: A = w + r
The triangle build by the touching-points
a´= 2 tA r / √( tA+r)=(b+c-a)r / √(¼(b+c-a)+r) etc.
It seems, that something still remains irrational
Proof for w = R² - d² (d is the distance of radii)
The bent of incircle is the sum of bents
of circles with the radii R+d and R-d .
A similar closure problem from J. Steiner
Condition for circle-quadrangles:
(R - d)-² + (R + d)-²= r-²
Cicle octagon

Part II: Quadrangles
The theorem of Wehrle for quadrangles
Wehrle for circle kites 2rR = abc / (a+b)
Wehrle for circle trapeziums 2rR = (a+c)√(a²+c²+6ac)
Rational quadrangles with integer sides

PART III: Pyramides
Still a dimension more
Just the volume is very simple
Thus the double product of radiu is
The Theorem of Fehringer-Wehrle

Part IV: n and infinite dimensions
From four to infinite dimensional space
Theorem of Pythagoras for n dimensions
Volume and surface of n-dim. spheres
Vanishing spheres

Appendix: The Theorems of F E H R I N G E R 113
Um- und Inkugelradien am allgemeinen Tetraeder
Erweiterung der Euklidischen Flächensätze auf das allgemeine Dreieck und die Fehringerschen Gleichungen nebst Anwendung zur Volumenbestimmung des allgemeinen Tetraeders.

References 144


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Portrait from Erich Meyer

The author, Hugo Wehrle, studied 1975-1982 mathematics and physics at the Albert-Ludwigs University Freiburg with the university state examination. Than he learned computer languages (programming: first COBOL, Assembler, Pascal, Unix, C) and later a course as systemadministrator for Windows- and Unix netwoks: the Microsoft certified system engineer (MCSC). Finally, after some other steps, I worked 1989-1996 in this IT-section by the AEG (general electric company) in Konstanz, producing postal sorting machines for the whole world; 1989 - 2001 by INTERNOLIX, producing software for online-jobs.

The author of the appendix is Arno Fehringer, who studies mathematics and physics at the Albert-Ludwigs University in Freiburg too, at about the same time as I did. He is still teacher for mathematics at the higher school (called „Gymnasiallehrer“) in Gailingen.

The subject of my book is elementary geometry in two, three, four and finally infinte dimensions. Brief examples of the content:

The area of a rectangular triangle is the sum of w and a quarter of w*

A = w + 1/4 w*

And there are some bradnew diagrams of Thoery of trigonomical Wehrle

The analogy of a right-angled triangle yields the three-dimensional theorem of Pythagoras ! If all branches at one edge of a polygonal pyramid are equal and the double of the height, than the radius of the circum sphere is just the length of these branches!

The circum radius R of any tetrahedron is formulated in the Theorem of Fehringer-Wehrle only by its branches:

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Than we leave the tree dimensional space, discuss four dimensional bodies and elevate finally to the simplices of infinite dimensional space, to show, that all the hypershperes have disappeared completely?

My book proves at the very end, that eucledean geometry can not be the last truth, or in other words, eucledean geometry is not the correct model for the really to describe the real space exactly, we live in[1] and physicists deal with! But nobody would ever notice this, if I would´nt write explicitely in the conclusion chapter. Without my logical conclusions, the book will be a work of elementary geometry for triangles, quadrangles, tetrahedrons and simplices, with some very new formulae (for example the one called Fehringer-Wehrle for general tetrahedrons), which never have been published before in any other book; last not least with a chapter of infinite dimensional eucledean space, where something strange happens: The spheres are diasappearing.

Purpose of the book is to present new ideas and to make people to be amazed! They shall be able, to come to the logical conclusion, that mathematics is reallity, but euclidean geometry is only a simplifying model. The book is a monograph and should be comprehensive for everybody, who really are interested in mathematics!

For certain, the reader must not have studied mathematics to understand, what I´m talkig about!


The book is a research monograph with many exercises at the end of the first three chapters. Research aereas are mew formulae for triangles, quadrangles and pyramides.

For example, that for any triangle the square of the relation of the incircle to circum circle radius is

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The radius of circum circle of a circle-trapezium is always c2/(2ab) times the length of the diagonals!

For right-angled bicentric quadrilateral kite with radius of incircle is

r = ac/(a+c), and of circum circle R = ½√(a²+c²) and distance of centers

|MuMi | = ½ {(r -x i) /(r+xi)} √(r²+xi²)

The distance of centers of the spheres of right-angled tetrahedrons is

|MiMu| = √[R2 + 3r2 - (a+b+c)r ] with the product of radii

4rR =[ab+ac+bc-√(a²b²+a²c²+b²c²)]√(a²+b² +c²) / (a+b+c)

For general tetrahedrons the radii product is

8rR = √ [(ad+be+cf) (-ad+be+cf) (ad-be+cf) (ad+be-cf)] / O

Why is this subject area important ?

Mathematics isn´t important for most people, only “things that nobody needs”! But we live in a contradictionary world. People use the technics, based in constructions, using the knowledge of mathematics, for example to look TV, or hear DVDs, or even to fly to the moon (IMPOSSIBLE WITHOUT COPMUTERS; thus impossible without mathematics of gravitation etc). But almost nobody cares and it isn`t honored, doing mahematics and searching for the very basic and ultimate reasons of the things, that keep the univers together, rather than playing football, baseball, basketball, rugby, tennis, beachball, soccer or other sports, or playing songs, where you can earn millions of dollars, standing on the stage or producing movies like R. Reagan (or Arnold Schwarzenegger), getting well-known by this way and therefore president.(or governor).

But it is very importand for western world, to be at the top in knowledge and technical know-how, to use newest machinery for the production of export articles, because otherwise the third world will overrun us! And mathematics is the basic language of all science and technology.

But I needn`t explain you the necessity of mathematics or geometry, or should I? My book is very good illustrated and comprehensible, it has colours and delights, a storehouse of new ideas, a productive ground for a lot of rosy geometical trees!

The best available book for elementary geometry is Donald COXETER`s >>Introduction to Geometry<<. For more philosophical questions like the question of Euklid reality for example or for metamathematics, it is, - as far as I know -, W. N. Molodschi`s >> Studien zu philosophischen Problemen der Mathematik << VEB Deutscher Verlag der Wissenschaften, Berlin 1977.


Can you imagine, that there are elementary and still unrevealed theorems for the triangle, although mathematicians handle with these most simple objects of planimetry since over 2 500 years?

Remember all the greek geometrical genius like the first philosopher Thales of Milet (≈625-547 BC), the vegetarian school of the brotherhood of Pythagoras (born at about 600 BC), the thirteen volumes of the elements of Euclid (living in Alexandria about 300 BC), Zenon from Elea (≈490-430) or Archimedes from Syracus (≈285-212)! The world-outlookings Aristarchos from Samos and Plolemäus from Alexandria. And finally Hypatia from Alexandria: her murderers have killed the mathematics too - for about thousand years, the mathematics disappears with her completely: The roman empire could not live without soldiers, but without mathematicians and this longer as any other on the world!

All these knew triangles, quadrangles and tetrahedrons! Could it be, that there is something still unfound, unsolved or not to find in any book of mathematics or formulary? You know, that the product of the sides of any triangle divided through their sum (called cicumference) is just the double product of their radius of the circumscribed and inscribed circles? That the sum of the sides of catheti is just the sum of the diameters: a+b = 2(r+R)? Their half product (the area) is the sum of the Wehrle-number w and a quarter of the Wehrle-number of the differnces w*:

A = w + 1/4 w*

Where w* is the square of the diameter of the incircle (2r) power 2.

The diameter of the incircle is just the sum of the two minor sides subtracted by the largest one. Can you express the right-angled sides only by their in- and circumscribed radius? You know the smallest of the rational (not rectangled) triangles, having natural numbers for the sides[2] ; or the smallest rational isosceles triangles with integer lengths of sides? Both can be constructed by using “pythagorean[3] ” triangles only

During the matter in chapter I are triangles, it`s quadrangles in chapter II. Witch of the quadrangles have an in circle or circum circle and what´s the product of their radii 2rR? Witch kind of quadrangles exists, having integer sides and areas, altitudes and in- or circum circles all rational?

You dont know the theorems of the Sinus-Wehrle or the Sinus-Wehrle of the differences, indicating the relation of the radii of the incircle to the circum circle of any triangle?

Than in chapter III for pyramides, we leave the plane and therefore we need one more dimension. The analogy of a right-angled triangles yields the right angled tetrahedron and the three-dimensional Theorem of Pythagoras ! You surely dont know the radius product of the in- and circum sphere of any tetrahedron deduced only by its branches?

If all branches at top vertice of any polygon-pyramide are equal, than its radius of circum sphere is just the square of this length divided through the double altitude of pyramide: Specially if the altitude is just half of length l of equal branches at the top, its circum sphere radius is just R=l!

You know, that the content of a sphere in any dimension relates to its boundary always like the dimension n to its radius r.

In the last chapter we leave the tree dimensional space and finally elevate to infinite dimensional space. Would you last not least understand, why the hypershperes there have vanished all completely? Everybody will think intuitively, that this is just impossible! That there exists no points of a constant distance to a fixed point, - the middle -, nobody can imagine! So, only one conclusion is possible: Something must be wrong with the euclidean geometry! That’s the reason why Albert Einstein, who tried decades of his last years to get the G and U nification T heory (GUT or TOE: Theory Of almost Everything), excused his failing with this statement:

“A new mathematics would be necessary”!

In this book, the basic problem about the number of dimensions of euclidean spaces up to the infinity of dimensions is approached. You must know, that there exist proofs in mathematics, that are only nominal. For example, almost all mathematicians will tell you, that there is much more than the infinity. They prove you, that there are much, much more real numbers (א 1) as integers (א 0).
But indeed, it´s infinite, and how can something be more as this potential and never reachable infinity, only existing in mind, but not real?

Yes, they will make you believe, that there is very, very much, much more and even infinite infinitely more as the infinity of the integer numbers: that there is no pope among the cardinals of infinities, this never ending infinity of infinities. They call this transfinite mathematics, and they handle with this infinities א n as like as with integers and, - bringing their own mindgames with them -, asking for the infinity א ½ according to the fraction ½, the midst between the first integers zero and unit! A cardinal error, because they will never ever find or prove it!

It might be, if there is a space with more than “only” one infinity of dimensions, - they could explain to you -, there could be two or more limiting values to infinity (for example for the volumes of n-dimensional unit-spheres).

Finally they make you sure to find statements, you never will be able to say, if it`s true or not (like Kurt Gödel does), anyway in wich sytem of axioms you will move. They tried to breaking down the whole mathematical building of causality! And I‘m sure, you will never, nerver really understand.

I will call a spade a spade! You will find no ambiguity at all, but only brilliant clarity. Perhaps you will suddenly understand mathematical statements, you never would dream of doing!

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Arthur Schopenhauer

Part I The Theorems of >>W E H R L E<<

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What have all these triangles common

The Wehrle-number is simple the product divided through the sum:

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For example, let’s take the first three natural numbers 1, 2 and 3.

Its Wehrle-number is w (1, 2, 3) = 1 x 2 x 3 : (1+2+3) = 1

Add now respectively two of these first natural numbers 1, 2 and 3, than you get as sums the three sides of a triangle with a =1+2 = 3, b =1+3 = 4 and c =2+3= 5. In this case it is even right angled, that means it has an angle of 90° degrees, because of the Theorem of Pythagoras: 3+4=5.

Its incircle-radius r is just this number of Wehrle 1.

(In general r is the square root of the Wehrle-number of these tangents!)

The Wehrle-number of the sides 3, 4 and 5 of this pythagorean triangle is

w (3, 4, 5) = 3 x 4 x 5 : ((4-1) + 4 + (4+1)) = 3 4 5 : (3 4) = 5

and all triangles of the figure above have the same Wehrle-number five !

In other words, all those triangles have the same product of the radius of incircle r and circum circle R.

You know, the Wehrle-number of the sides of a triangle, (shorter: of the triangle) is just the double product of the two radius, that is of the incircle, touching all sides, and of the circum circle, passing through all three edges. (Two points define a straight line uniquely, and three a circle, which center is the crossing point of the perpendicular bisctors of the sides defined by these three edges!)

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Example: 3 x 4 x 5 : (3 + 4 + 5) = 2 x 1 x 2,5 where r = 1 and R = 2,5

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Tangents are 1, 2 and 3

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The area of w and w* (a quater of w* is r 2 )

Cause the Wehrle-number has the unit of an area, it could be visualized by an area too. Now, the Wehrle-number of an rectangular triangle is just the area of the triangle, subtracting the quarter of another Wehrle-number, the Differences-Wehrle w*:

A = w + ¼ w* .

Because this Wehrle-number of differences w* is exactly the square of the diameter of the incircle, as we will see later, it follows:

w = A - r 2

The area of rectangular triangles is the sum of the Wehrle-number and the square of the incicrle-radius!

Example: A = ½ x 3 x 4 = 6 and because r = 1, it is w = 6 - 1 = 5

The double area 2A is just the area of a rectangle with the sides, joining the right angle. Therefore

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and the product of the two catheti is with w=2rR logical

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Example: For a=3 and b=4 follows r=1 and R=2,5: 3x4 = 2x(5+1)

The largest side of a rectangular triangle is called hypotenuse. Because the longest side opposites the largest angel, the smallest side opposites the smallest angel, the hypotenuse opposites always the greatest right angle (The sum of angles is 180°, if the postulate of parallels is valid)

Let the hypotenuse be c. This is the diameter of the circum circle too, and we will understand soon, that the diameter of the incircle is exactly

2 r = a+b – c ( = differences of sides dc)

The roundabout way over the two catheti a and b is exactly 2r more (the diameter of the incircle) than the direct way by hypotenuse c.

Cause the hypotenuse is the diameter of the circum circle- (circle of Thales), we get

The sum of catheti is just the sum of diameters!

a+b = 2 (R+r)

Example: a=3 and b=4 a+b = 3+4 = 5+2 = 2R+2r

Both equations have the two variables a and b containing only terms with the radius r and R and yield together a theorem for the catheti a1 and a2 only depending from the radii:

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where a = a1 and b = a2 , or vis versa

Example: R=2,5 and r=1 yields for the sum of radii R+r = 2,5+1 = 3,5

The term under then radical sign is 2,5 - 1 - 5 = 0,25 a square here, their root 0,5. Hence the catheti are a= 3,5 +0,5 =4 and b = 3,5 - 0,5 = 3 (hypotenuse is 2R=5).

Back to general triangle. Let’s call these expressions of three sides, where from the sum of two the third is to be subtracted

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as differences.

These are precisely the double tangents, in which the touching point of incircle divides the side. The very small one is the radius of the incircle, cause the hypotenuse c is the largest side, opposite the right angle (at the opposite edge C the tangents form a square).

These differences express the so-called „un-equalities of the triangle“, that means, that the sum of two sides has always to be greater than the third side:

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therefore dc = a+b-c has to be positiv etc.

Construct now the Wehrle-number of the differences w * („w-star“),

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where the denominator is

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We result the square, surrounding the incircle, for any triangle

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The largest side opposing edge and both touching points of the together with the incircle-center Mi form a square, and the Wehrle-number of the differences w * is the circumscribed square of the incircle 4r2 .

The smalles rational triangles with integer sides

If the Wehrle-number of tangents is a square, we can construct a rational or discreet triangle (no square roots to count), that means with rational area, altitudes, sinus values of the angles and radii too. If the sides of the triangle shall be natural (integer), than the smallest I proved, which is not rectangular, has the tangents 1, 3 and 12 with the Wehrle-number w(1,3,12) = (1 3 12) : (1+3+12)= (3:2)2.

Its radius for the incircle is therefore 1,5.

The sides 4, 13 and 15 are the three two-sums of its tangents 1, 3 and 12:

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r=1 x 3 x 12/(1+3+12)=9/4, thus r= 1,5 Its radius of circum circle is R=w/2r= 4x13x15 /(4+13+15) :3 = 65 : 8 = 8,125

and its area is A = ½ru =½1,5x32 = 24

The tangents x, y and z building a Wehrle-square-number are

x= l - m2 y = l (l - m2) and z =m2 (l +1)

with rational l and m ( l > m2 )

Example: For l=3 and m=1,5 you receive x=3:4 , y=9:4 and z=9.

Multiplied by 4:3 results in x=1, y=3 and z=12, the discrete triangle with sides a=4, b=13 and c=15 and r=1,5.

Specially for m=1 you get only rectangular triangles:

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Example: l=2 yields the minimal discrete one a=3, b=4 and c=5.

The general pythagorean triads are: l2 - m2 , 2lm und l2+ m2

The resulting discrete triangle has the following sides:

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Example: For l=5 and m=2: a = 6, b = 25 and c = 29 with r = 2, A=60.

Every rational triangle consists of two rational rectangular ones!

If it`s a right angled one, the relation of the parts of hypotenuse c devided by the altitude is the relation of squares of cathti: q.p= a : b

Because the area is rational, the altitude (2A/side) is rational too! If one or both area(s) of the rectangular parts is (are) irrational, the sum of the areas A would not be rational. Therefore the sides in question of the rectangular triangles can not be (pure) irrational.

Example: The triangle with the sides 4, 13 and 15 can be split up into the two rectangular with the sides, 2,4, 3,2 and 4 and the other with sides 3,2, 12,6 and 13., The largest side 15 is divided by the height (altitude) into 2,4 and 12,6

Counter- Example: The triangle 2, 5 and 6 has the area

¼√(2+5+6)(-2+5+6)(2-5+6)(2+5-6) = ¼√(13 x 9 x 3 x1) = ¾√39.

The largest side 6 is divided by its altitude into 1,25 and 4,75. But the altitude is not rational: h = ¼√39

The distance of the centers of in- and circum circle

Very interesting is distance of the centers Mi and Mu, we will note as d = |MiMu|. In rectangular triangle d is always greater the radius r of incircle, cause its the altitude over the hypotenuse c of the triangle, perpendicular to c like the radius of incircle. The middle of c yields Mu and the radius of the circum circle is R= ½. The sides are connected with d through the equation

2(a+b) c = 3c – 4d. As we will still see, this distance d of the center Mi and Mu is not really connected with the sides of the triangle, - and therefore depends not from the concrete triangle itself-, but is only connected with the radii r and R.

You can not construct any triangle into too circles, the one containing the other, without touching it, but only for a certain determined distance of centers. This distance d of centers is very important and has to be he geometrical average of the bigger radius R and the difference of R - 2r:

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Therefore the diameter of the incircle 2r has to be smaller than the radius of the circum circle R, and is at the outside equal, that is only for he regular case, the equilateral triangle, where the radius of circum circle R is just equal to the diameter of the incircle 2r. Because here the perpendiculars in the middle of the sides divide the angles too, and cut each other in the relation 1:2. Cause of R=2r it results a zero distance d=0: Like for every regular polygon, its in- and circum circle are concentrically.

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For two circles in one another there is

either not a single triangle at all

or there exists an infinite number of triangles !

In case of rational radii and distance of centers, the sides are irrational

(here the catheti are 11+√7 and 11-√7)

The distance condition can be formulated too as:

The square of the distance d has to be the difference of the square of circumradius R and the Wehrle-number!

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Comparison of the differences

w = A - r  with w = R– d  (broken lines)

In this special case, there are the cuts of the hypotenuse 2 and 3

with just the product h = A

For rectangular triangles the distance d of the centers of in- and circum circle is always greater than the radius of the incircle r, cause r and d form a rectangular triangle at the hypotenuse c, except for the symmetrical case of an isosceles right angled triangle, - half a square -, where d is perpendicular to the hypotenuse and d = r= a-½c=½ (√2 -1) c.

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The area of a rectangular triangle (half a rectangle) is smaller than the square of circumradius; equality A = R only for the isosceles right angled one (half a square)

The distance d of centers for right-angled triangles is:

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Proving the theorem of WEHRLE!

The Wehrle-number w of the three sides a, b and c of a triangle, is the double product of the radii of the incenter r and circumcenter R

abc : (a+b+c) = 2 rR

and the square root of the differences-Wehrle-number w « of the sides dc = a+b - c , db = a b + c and da = - a + b + c is equal to the incircle-diameter

√ w*= 2r

As you all know, every triangle is definite determined by the length of its three sides, that is, its appearance or form is uniquely described. Therefore all things of the triangle can be calculated by these three sides!

For example the bisector of the side a is[illustration not visible in this excerpt],

the bisector of the angle at he edge A [illustration not visible in this excerpt] : (b+c) or the altitude through A is

[illustration not visible in this excerpt] , which is longer than the incircle-radius[illustration not visible in this excerpt] for the amount of r(b+c):a

The square of a side ca be computed by the square-sum of the other ones, corrected by subtracting the double product of these sides and the cosines of thereby defined angle c2 = a2 +b2 - 2 a b cos γ.. Therefore is

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On the other side, you know sin2 γ + cos2 γ = 1, and sin γ is calculated as:

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and finally

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II.) This sinus-value you can calculate too as sin γ = c : (2R) Equate I. with II. yields

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Therefore the product of the three sides is

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This is proved by simple calculation:

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Because the sum of differences is u = (-a+b+c)+(a-b+c)+(a+b-c), it follows:

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It remains to show, that √w* = 2r .

With I.) put into A = ½ ab sin γ, and on the other hand A = ½ (a+b+c) r

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is now proved:

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The sum of the three sinusvalues of the angles of any triangle (left)

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and the sum of the squares of the three sinusvalues z = sin αi of the angles (right)

for the angles x and y from 0 to π

The sum of the sinus-values sin αi is proportional to the sum of the sides ai and the sum of the sinus-squares is is proportional to the sum of the squares of sides. General

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[1] As we all know, Albert Einstein proved, that the space is not even. But telling this to a working man, that space has a curvature, than he closes the doors suddenly („die Rolläden gehen herunter und der Bordstein wird hochgeklappt“)! And this will be the end of every discussion, with the result, that you will be in his eyes a lunatic all live long!

[2] It`s integer sides are 4, 13 and 15 The thre integer sides are the pythagorean triplets: l2 - m2 , 2lm und l2+ m2 for integer l and m



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Titel: Beitrag zum Jahr der Mathematik 2008