WHAT IS THE pitagoras theoreme?
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Don't you mean the Pythagorean Theorem?  That theorem is a basic tenent of Geometry that states: In any right triangle, the square of the length of the hypotenuse (side opposite the right angle) is equal to the sum of the squares of the other two sides.   A^2 + B^2 = C^2.
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you can't go wrong with :\n\nThe quadratic formula\nThe Pythagorean theorem\nlaw of sin/ law of cos (for trig)\narea and circumference of a circle\n\nThese are some general formulas that appear on exams\nI could be more precise if i knew the level you are aiming for.
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good question... what thereoem are you talking about?  Chebychev invented more than one theorems.
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This is an easy problem.  Here are the rough ingredients:\n\n(1) f is an isometry (i.e. d(f(x),f(y)) = d(x,y) for all x,y in X) implies that f is continuous.  (Prove this!)\n\n(2) The image of a compact set under a continuous map is a compact set, and so f(X) is a compact subset of X.\n\n(3) Suppose, for contradiction, that f is not onto.  So there is x in X &#92; f(X).  Let epsilon be positive real number less than dist(x,f(X)) (this distance is defined to be the infimum of the distances d(x,y) for all y in f(X)).  Since X is compact, X can be cover by finitely many balls of radius epsilon.  Let N be the smallest number of balls of radius epsilon with which you are able to cover X.  x belongs to one of these balls, obviously, and such a ball is disjoint from f(X).  So there is a cover of f(X) consisting of at most N-1 balls of radius epsilon.  But the inverse images (under f) of these balls (all of them of radius epsilon because f is an isometry) cover X.  Since we chose N to be minimal, and we have found now an epsilon-cover of X consisting of N-1 balls, we have reached a contradiction.
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The formula you give is called Heron's Theorem, which is technically a case of Brahmagupta's Theorem.  Brahmagupta's Theorem is for cyclic quadrilaterals, meaning ones that can be inscribed in circles.  The theorem is \nArea = sqrt ( (s-a)(s-b)(s-c)(s-d) )  where s is the semiperimeter and a,b,c,d are the sides.  When d=0, that gives you a triangle (actually there's an extra semiperimeter in your formula).  Anyway, Brahmagupta's only works for cyclics.\n\nFor non-cyclic quadrilaterals, there is no one formula.  For trapezoids the formula is A = 1/2*h*(b1 + b2) where b1 and b2 are the parallel bases, h is the distance between them.  For a kite, A = d1*d2 where those are the diagonals.  For any parallelogram, A = l*w (length and width).  For more irregular shapes, the best thing to do is split them up into triangles and work it out that way.\n\nHope that helps you!
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Finagle's principle:  The perversity of the universe tends toward a maximum.
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First, define a new function g(x)=f(x)-x. f(x) has a fixed point iff ∃x: g(x)=0. By the fundamental theorem of calculus:\ng(x) = g(0) + [0, x]∫g'(a) da\n\nLet c be sup(|f'(x)|). Then f'(x)≤c<1, and so g'(a)≤c-1<0. So for x>0:\ng(x) < g(0) + [0, x]∫c-1 da = g(0) + x(c-1).\nAnd for x<0:\ng(x) > g(0) + [0, x]∫c-1 da = g(0) + x(c-1).\n\nHowever, [x→∞]lim g(0) + x(c-1) = -∞ (since c-1 is negative), and since g(x) is bounded above by this function for x>0, [x→∞]lim g(x) must also be -∞. Conversely, [x→-∞]lim g(0) + x(c-1) = ∞, and since g(x) is bounded below by this functon for x<0, [x→-∞]lim g(x) is ∞. But this means that g(x) is positive for some negative value of x and negative for some positive value of x, thus by the intermediate value theorem, ∃x: g(x)=0 and therefore f(x) has a fixed point. To see that it's the only one, consider that if g(x) had two zeros then by Rolle's theorem there would exist a point where g'(x) is zero, which contradicts the fact that g'(x) is strictly negative everywhere.\n\nNote that this proof remains valid if c is merely the supremum of f'(x), instead of |f'(x)|.\n\nFor the counterexample with |f'(x)|<1 (but not sup(|f'(x)|)<1), consider f(x)={1/x+x if x≥1, 2 if x<1}. This function is everywhere differentiable ([x→1-]lim f(x) = [x→1+]lim f(x) = 2, and [x→1-]lim f'(x) = [x→1+]lim f'(x) = 0), and ∀x, |f'(x)|<1, but f(x) has no fixed points.
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www.cabrillo.edu/~cfigueroa/4A/4Aexamples/4Aexample9-F06.pdf
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Here is one condition:\n\nR1^2  +  R2^2  =  AB^2 (Pythagorean theorem),\n\nwhere R1 and R2  are the two intersecting circles radii and AB is the distance between their centers (AB < R1 + R2)
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I have a copy of the second edition. Is that good enough?\n\nIs there something in particular you need, or do you want the outline of the whole book?\n\n1. Set theory\nIntroduction\nFunctions\nUnions, Intersections, and complements\nAlgebra of sets\nThe axiom of chice and infinite direct product\nCountable sets\nRelations and equivalences\nPartial orderings and the maximal principle\nWell ordering and the countable ordinal\n\nPart One -- Theory of Functions of a Real Variable\n\n2. The Real Number System\n\nAxioms for the real numbers\nThe natural and rational numbers as subsets of R\nThe extended real numbers\nSequences of real numbers\nOpen and closed sets of real numbers\nContinuous functions\nBorel sets\n\n3. Lebesgue Measure\nIntroduction\nOuter measure\nMeasurable sets and Lebesgue measure\nA nonmeasurable set\nMeasurable functions\nLittlwood's three principles\n\n4. The Lebesgue Integral\nThe Riemann integral\nThe Lebesgue integral of a bounded function over a set of finite measure\nThe integral of a nonnegative function\nThe general Lebesque integral\nConvergence in measure\n\n5. Differentiation and Integration\nDifferentiation of a monotone function\nFunction of bounded variation\nDifferentiation of an integral\nAbsolute continuity\nConvex functions\n\n6. The Classical Banach Spaces\nThe Lp spaces\nThe Holder Minkowski inequalities\nConvergence and completeneww\nBounded linear functionsals on the Lp spaces\n\n\nPart Two -- Abstract Spaces\n\n7. Metric Spaces\nIntroduction\nOpen and closed sets\nContinuous function and homeomorphisms\nConvergence and completeness\nUniform continuity and uniformity\nSubspaces\nBaire category\n\n8. Topological Spaces\nFundamental notions\nBases and countability\nThe separation axioms and continuous real-valued functions\nProduct spaces\nConnectedness\nAbsolute (G delta)\nNets\n\n9. Compact Spaces\nBasic properties\nCountable compactness and the Bolzano-Weierstrass property\nCompact metric spaces\nProducts of compact spaces\nLocally compact spaces\nThe Stone-Cech compactification\nThe Stone-Weierstrass theorem\nThe Ascoli theorem\n\n10. Banach Spaces\nIntroduction\nLinear Operators\nLinear Functionals and the Hahn-Banach theorem\nThe closed graph theorem\nTopological vector spaces\nWeak topologies\nConvexity\nHilbert space\n\nPart Three -- General Measure and Integration Theory\n\n11. Measure and Integration\nMeasure spaces\nMeasurable functions\nIntegration\nGeneral convergence theorems\nSigned measures\nThe Radon-Nikodym theorem\nThe Lp spaces\n\n12. Measure and Outer Measure\nOuter measure and measurability\nThe extension theorem\nThe Lebesgue-Stieltjes integral\nProduct measures\nInner measure\nExtension by sets of measure zero\nCaratheodory outer measure\n\n13. The Daniell Integral\nIntroduction\nThe extension theorem\nUniqueness\nMeasureability and measure\n\n14. Measure and Topology\nBaire sets and Borel sets\nPositive linear functionals and Baire measure\nBounded linear fuctionals on C(X)\nThe Borel extension of a measure\n\n15. Mappings of Measure Spaces\nPoint mappings and set mappings\nMeasure algebras\nBorel equivalences\nSet mappings and point mappings ofn complete metric spaces\nThe isometries of Lp
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I know of two:\n\nTranslation and rotation: \n\n1) translation :\n\n Take a rectangle and cut a diagonal and extended the lines  from all three side of the triangle at right angle to it  from the side that is made from cutting the rectangle until those lines equal the lenght of the side of the triangle drawn from  and it  form three boxes  around all three sides of the triangle. Now let take the big side an calulate the square  of it ,then calulate the squares of the other two sides and add themup,  it should equal the square of the big side and that proves the pythagorean theorem. \n\nRotation: \n\nSee the show : Project Mathematics by Cal tech : Entitled "The Pythagorean Theorem" it explain better  than I can, It worth watching. Very good show on Math & Geometry
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You might try reading Whitehead and Russell's Principia Mathematica.  I can't promise it will help, as I haven't read it, but I've been told that this issue is addressed.
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ℤ   \nℤ⁺  \nℤ* \nℤ-action\nℤ-action of C^r\nℤ₁ - extension\nZ (Zermelo set theory)\nZF (Zermelo-Fraenkel set theory)\nz-axis  \nz-coordinate  \nz-distribution\nz-plane\nZ-score\nZ-set\nz-sphere\nz-transformation\nZariski closed (set)\nZariski connectedness theorem\nZariski decomposition\nZariski dense (set)\nZariski main theorem\nZariski open (set)\nZariski ring\nZariski site\nZariski topology (of a spectrum, of a variety)\nZassenhaus group\nZassenhaus theorem\nZeeman unknotting theorem\nZeller theorm\nzenith\nzenith distance (ζ)\nzenith angle\nZeno's paradoxes\nZermelo-Fraenkel set theory\nZermelo set theory\nzero(s)\nzero algebra\nzero angle\nzero cycles\nzero-dimensional space (Baire)\nzero divisor\nzero element\nzero homomorphism\nzero matrix\nzero object\nzero-one law\nzero point\nzero-point theorem\nzero representation\nzero ring\nzero section\nzero-sum game\nzeroth law of thermodynamics\nzero vector\nZFC\nZilber theorem\nzeta function\nzonal harmonic\nzonal polynomial\nzonal spherical function\nzone\nZorn lemma\nZygmund class\nZygmund singular integral operator\nZygmund type
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So you need to prove that each element (x,y) E br(xo, yo) has a neighbourhood contained in br(xo, yo),\n\nTake a point (xa,ya) E br(xo,yo) and let d = r - ||(xa,ya) - (xo,yo)||.  d is positive since ||(xa,ya) - (xo,yo)|| < r.\nThen the neighbourhood bd(xa,ya) satisfies the condition because for any point (x,y) in bd(xa,ya) we have\n||(x,y) - (xo,yo)||\n<= ||(x,y) - (xa,ya)|| + ||(xa,ya) - (xo,yo)|| (triangle inequality)\n< d + ||(xa,ya) - (xo,yo)|| (because (x,y) E bd(xa,ya))\n= r.\n\nIn plain English: If the point (xa,ya) has distance d from the boundary of the set, then a neighbouroud of (xa,ya) whose elements are all less than d away from (xa,ya) will be contained in the set as well.
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Fundamental Theorem of Linear Programming:\n\nIf the constraints define a feasible, bounded convex polyhedron, then the maxima and minima are attained on the boundary points of that polyhedron.\n\nFor an example, check out http://people.brunel.ac.uk/~mastjjb/jeb/or/morelp.html
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The  Pythagorean Theorem gives the length of the hypotenuse (the longest side) of a right triangle ( one of the angles is 90 degrees )\na^2 + b^2 = c^2\n\n\nApplying the Pythagorean Theorem twice you get a three dimensional version \n\na=x\nb=y\nx^2 + y^2 = c^2\nc^2 + z^2 = h^2 (using c as the side of a new triangle)\nx^2 + y^2 + z^2 = h^2\n\nwhich is the length of any line from (0,0,0) to (x,y,z)\n\nthe distance formula just finds the change in x, y, and z and puts it into this formula\nx = x1-x2\ny = y1-y2\nz = z1-z2\n\nThe midpoint formula is really unrelated to the other two.\nAll it does is average each of the coordinates of the other two points.  ( (x1+x2)/2 , (y1+y2)/2 , (z1+z2)/2 )\nThe only thing it has in common is the use of the same type of coordinate system
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One. \n\nThis relation is really just Pythagoras' theorem, hiding in a new form. Recall:\n\nH^2 = A^2 + B^2 \n\nThat is, the square of the hypotenuse H is the sum of the squares of the other two sides A and B, for a right triangle with sides A, B and hypotenuse H.  \n\nOf course, A=H*cosx and B=H*sinx for some angle x, so that \n\nH^2= (H^2)*(sinx*sinx + cosx*cosx)\n\nleading to the result:\n\n1=sinx*sinx + cosx*cosx\n\nThis result also is nicely connected to the simple fact that \n\nx - x = 0 \n\nbecause \n\nexp(i*x)=cosx + i*sinx and\nexp(-i*x)=cosx -i*sinx\n\nwhere i=sqrt(-1). \n\nThen multiplying these two, \n\nsinx*sinx + cosx*cosx=exp(i*x)*exp(-i*x)=\nexp(i*[x-x])=exp(0)=1