Pythagorean theorem
The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides, expressed as a^2 + b^2 = c^2, where c is the hypotenuse and a and b are the legs.[1] This relation holds exclusively for right triangles and forms a cornerstone of Euclidean geometry, enabling the calculation of unknown side lengths when the other two are known.[2] Although commonly attributed to the ancient Greek philosopher and mathematician Pythagoras (c. 569–500 BCE), the theorem was known much earlier to civilizations such as the Babylonians and Egyptians around 1900 BCE, as evidenced by the Plimpton 322 clay tablet, which lists numerous Pythagorean triples (integer solutions to the equation).[3] The Egyptians applied it practically, using ropes knotted in a 3-4-5 ratio to form right angles for construction and surveying.[4] Pythagoras and his followers in the Brotherhood of Pythagoreans formalized geometric proofs of the theorem, treating mathematics as a mystical pursuit, though they kept discoveries secret; a proof later appeared in Euclid's Elements (c. 300 BCE) as Proposition I.47.[4] The theorem's discovery reportedly led the Pythagoreans to confront irrational numbers like \sqrt{2}, challenging their belief in numerical harmony.[4] Over centuries, the theorem has inspired hundreds of distinct proofs, with Elisha Scott Loomis's 1940 compilation documenting 370, including geometric dissections, algebraic derivations, and similarity-based arguments from figures like Euclid, Bhāskara II, and even U.S. President James Garfield.[5] Its converse—if a^2 + b^2 = c^2, then the triangle is right-angled—further extends its utility for verifying angles.[2] The theorem's importance extends beyond pure mathematics into practical applications, serving as the basis for defining distance in Euclidean space (e.g., the distance formula \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2})[6] and enabling solutions in fields like architecture, navigation, physics, and engineering.[7] For instance, archaeologists use it to lay out precise grids on excavation sites,[3] while surveyors apply it to measure inaccessible distances.[8] In modern contexts, it underpins GPS calculations[9] and computer graphics for rendering 3D models.[10]Statement and Basic Properties
Formal Statement
In a right-angled triangle, the square on the side opposite the right angle is equal to the sum of the squares on the sides containing the right angle.[11] This is formally stated in Euclid's Elements (Book I, Proposition 47) as the foundational relation for Euclidean geometry.[12] The two sides enclosing the right angle, known as the legs, are typically denoted by lengths a and b, while the side opposite the right angle, called the hypotenuse, has length c.[11] The theorem thus expresses the equation a^2 + b^2 = c^2, where a > 0, b > 0, and c > 0 are positive real numbers satisfying the conditions of a right-angled triangle in the Euclidean plane. This relation is independent of the units of measurement, as it derives from the geometric properties of lengths rather than specific scales.Geometric Illustration
The Pythagorean theorem is geometrically illustrated using a right-angled triangle, where the right angle is marked at one vertex, and the two legs are labeled as sides of lengths a and b, while the hypotenuse opposite the right angle is labeled c.[11] A simple numerical example is the right triangle with legs of lengths 3 and 4 units and hypotenuse of length 5 units, which verifies the relation through $3^2 + 4^2 = 9 + 16 = 25 = 5^2.[13] To visualize the theorem, squares are constructed outwardly on each side of the triangle: a square of side a with area a^2, a square of side b with area b^2, and a square of side c with area c^2.[14] Intuitively, this arrangement demonstrates that the combined area of the squares on the legs equals the area of the square on the hypotenuse, providing a visual basis for the theorem's core idea.[14]Proofs
Proofs Using Similar Triangles
One common geometric proof of the Pythagorean theorem relies on the properties of similar triangles, which arise from dropping a perpendicular from the right angle to the hypotenuse in a right-angled triangle.[15] Consider a right-angled triangle \triangle ABC with the right angle at B, legs AB = c and BC = a, and hypotenuse AC = b. Construct the altitude from B to the hypotenuse AC, meeting AC at point D. This altitude divides \triangle ABC into two smaller right-angled triangles: \triangle ABD and \triangle CBD, both sharing the right angle at D.[15] These smaller triangles are similar to each other and to the original triangle \triangle ABC by the AA (angle-angle) similarity criterion. Specifically, \triangle ABC \sim \triangle ABD because both have a right angle and share \angle BAC. Similarly, \triangle ABC \sim \triangle CBD because both have a right angle and share \angle BCA. The segments of the hypotenuse are AD = p and DC = q, with p + q = b.[15] From the similarity \triangle ABC \sim \triangle ABD, the ratios of corresponding sides yield \frac{c}{b} = \frac{p}{c}, so c^2 = b p. From \triangle ABC \sim \triangle CBD, \frac{a}{b} = \frac{q}{a}, so a^2 = b q. Adding these equations gives a^2 + c^2 = b(p + q) = b \cdot b = b^2, proving the theorem.[15] A variant of this approach, known as Garfield's proof, incorporates similar triangles within a trapezoid construction to derive the theorem via area comparisons. Developed by James A. Garfield in 1876 and published in the New-England Journal of Education, the proof arranges two copies of the original right triangle with legs a and b alongside a segment connecting their non-adjacent legs, forming a trapezoid with parallel sides of lengths a and b, and height a + b.[16] The area of this trapezoid equals \frac{1}{2}(a + b)(a + b) = \frac{a^2 + 2ab + b^2}{2}. Alternatively, the trapezoid decomposes into the two right triangles (each with area \frac{1}{2}ab) and an inscribed isosceles triangle with sides c, c, a + b (area \frac{1}{2}c^2), yielding total area ab + \frac{1}{2}c^2. Equating the areas simplifies to a^2 + b^2 = c^2. This method implicitly leverages the geometric relations akin to similar triangles through the shared altitudes and proportions in the figure.[16] These proofs assume Euclidean geometry, where parallel lines and perpendiculars behave as defined, and rely on the basic proportionality theorem for similar figures, ensuring corresponding sides are proportional.[15]Algebraic Proofs
One common algebraic proof of the Pythagorean theorem places a right triangle on the Cartesian coordinate plane for straightforward computation of side lengths using the distance formula.[17] Consider a right triangle with the right angle at the origin, one leg along the x-axis from (0,0) to (a,0), and the other leg along the y-axis from (0,0) to (0,b), where a and b are the lengths of the legs.[17] The hypotenuse connects (a,0) to (0,b). The distance between these endpoints, denoted c, is calculated as follows: c = \sqrt{(a - 0)^2 + (0 - b)^2} = \sqrt{a^2 + b^2}. Squaring both sides yields c^2 = a^2 + b^2, which is the Pythagorean theorem.[17] This approach generalizes to any right triangle by assigning arbitrary coordinates to the vertices and applying the Euclidean distance formula between points, confirming the relation holds regardless of placement.[17] For vertices A=(s,t), B=(u,v), and C=(m,n) with the right angle at C, the squared distances satisfy a^2 + b^2 = c^2 after expansion and cancellation of terms, leveraging the perpendicularity condition that the product of the slopes of legs AC and BC equals -1.[17] An equivalent vector-based algebraic proof uses the dot product to express perpendicularity and magnitudes. Let \vec{u} and \vec{v} be vectors representing the legs of the right triangle, so their dot product \vec{u} \cdot \vec{v} = 0 since they are orthogonal. The hypotenuse corresponds to the vector \vec{u} + \vec{v}, and its magnitude squared is: |\vec{u} + \vec{v}|^2 = (\vec{u} + \vec{v}) \cdot (\vec{u} + \vec{v}) = |\vec{u}|^2 + 2(\vec{u} \cdot \vec{v}) + |\vec{v}|^2. Substituting the orthogonality condition \vec{u} \cdot \vec{v} = 0 simplifies to |\vec{u} + \vec{v}|^2 = |\vec{u}|^2 + |\vec{v}|^2, or c^2 = a^2 + b^2.[18] This formulation extends naturally to higher-dimensional Euclidean spaces.[18]Dissection and Rearrangement Proofs
Dissection and rearrangement proofs of the Pythagorean theorem demonstrate the equality a^2 + b^2 = c^2 by cutting geometric figures into pieces and reassembling them to show that the total area remains unchanged, relying solely on the invariance of area under such transformations. These proofs date back to ancient China, with examples in texts like the Zhoubi Suanjing (c. 200 BCE) and later refinements by Liu Hui (3rd century CE).[19][20] They gained prominence in medieval Indian mathematics, and were further refined in the 19th century with more intricate cuttings, emphasizing the theorem's geometric essence through tangible rearrangements.[21] One of the most elegant and historically significant dissection proofs is attributed to the 12th-century Indian mathematician Bhāskara II, presented in his treatise Lilavati around 1150 CE. In this proof, four copies of the right triangle with legs a and b and hypotenuse c are arranged with their hypotenuses facing outward to form the boundary of a large square of side length a + b. This arrangement leaves an inner square tilted at an angle, with side length c, as the hypotenuses align to create its edges. The area of the large square equals the area of the inner square plus the areas of the four triangles: (a + b)^2 = c^2 + 4 \cdot \frac{1}{2}ab, which simplifies to a^2 + b^2 = c^2 upon expansion and cancellation. Bhāskara provided no algebraic steps, instead accompanying the diagram with the single word "Behold!" to invite visual verification of the area equality through the non-overlapping, gap-free tiling. This proof, while primarily an arrangement, functions as a rearrangement by conceptually shifting the triangles to fill the spaces around the inner square, highlighting the theorem without words.[1][22] A more explicit dissection proof, involving actual cuts and reassembly, was developed by British stockbroker and amateur mathematician Henry Perigal in the early 19th century and published in 1873. Perigal's method dissects the squares built on the legs a and b into pieces that can be rearranged to exactly fill the square on the hypotenuse c, proving the area equality directly. To achieve this, start with the square on leg b (assuming b > a); locate its center and draw lines parallel and perpendicular to the hypotenuse c, dividing the square into five pieces. Meanwhile, the square on c is dissected by connecting midpoints of its sides with lines parallel to the legs a and b, forming a central quadrilateral congruent to the square on a and four corner triangles. The key rearrangement involves transposing the five pieces from the square on b to match and fill the four corner regions and remaining areas of the c^2 square, while the intact square on a fits into the central area without overlap or gaps. This five-piece dissection visually confirms that the combined areas of the leg squares equal the hypotenuse square, with the symmetry aiding in the intuitive understanding of the fit. Perigal, who spent over 40 years exploring such geometric puzzles, engraved this diagram on his tombstone, underscoring its personal significance.[21][23][24]Euclid's Proof
Euclid's proof of the Pythagorean theorem appears as Proposition 47 in Book I of the Elements, establishing the theorem using only the axioms and previously proven propositions up to that point, without relying on the concept of similar triangles introduced later in Book VI.[25] The proof proceeds by constructing squares outwardly on each side of a right-angled triangle and demonstrating through a series of area equalities that the square on the hypotenuse equals the sum of the squares on the other two sides. This approach emphasizes geometric constructions and the properties of parallelograms, drawing on Propositions 41 through 46 of Book I, which deal with the construction of parallelograms, their equal areas when sharing the same base and height, and the equality of areas between certain triangles and parallelograms.[11][26] Consider a right-angled triangle ABC with the right angle at C, legs CA and CB, and hypotenuse AB. Construct square ABDE outwardly on the hypotenuse AB, square BCFG outwardly on leg CB, and square ACHK outwardly on leg CA.[25] Through point C, draw line CL parallel to the sides BD and EF of the square on AB (using Proposition I.31 for parallels). Join points AL and CK. The diagram thus features the triangle inscribed within these outward squares, with the parallel line CL creating parallelograms and auxiliary triangles that facilitate area comparisons; notably, the regions form gnomons, or L-shaped figures, added successively to build up the areas.[11][26] The proof unfolds in steps leveraging congruence and area properties. Parallelogram ACLM (with base AC and height equal to that of square ACHK) equals twice triangle ACK by Proposition I.41, and since square ACHK equals twice triangle ACK (as the square is composed of two such triangles), the parallelogram ACLM equals square ACHK. Similarly, for the other leg, parallelogram BLCN equals square BCFG through identical reasoning applied to triangle BCK and the square's composition.[11][26] These parallelograms together form the square ABDE minus the small triangles at the ends, but by adding the areas step-by-step—effectively using a shearing transformation via the parallels to equate regions—the entire square ABDE is shown equal to the sum of squares ACHK and BCFG (Propositions I.43 and I.45 ensure the outer figure's area equality under parallel shifts).[25] Thus, the square on the hypotenuse equals the sum of the squares on the legs, completing the proof under Euclid's axiomatic system.[11]Other Geometric Proofs
One notable geometric proof employs an area-preserving shear transformation, which slides shapes parallel to a fixed direction while maintaining their areas. In this approach, consider a right triangle with legs a and b, and hypotenuse c. The squares constructed on legs a and b are sheared into parallelograms that can be rearranged or transformed without altering their total area a^2 + b^2. Through successive shears and translations—operations that preserve base and height—these figures align to form the square on the hypotenuse, demonstrating that its area c^2 equals the sum, thus a^2 + b^2 = c^2. This method, attributed to Hermann Baravalle, highlights the invariance of area under affine transformations.[27] Another geometric proof, devised by Albert Einstein in his youth, relies on a mild dissection using auxiliary lines rather than full rearrangement. Start with a right triangle ABC where \angle C = 90^\circ, legs a and b along CA and CB, and hypotenuse c along AB. Drop a perpendicular from C to AB, meeting at D, which divides the original triangle into two smaller right triangles similar to each other and to the original. Construct additional similar triangles on sides a and b: one with hypotenuse b and the other with hypotenuse a. These auxiliary triangles, when positioned appropriately, dissect the original triangle into regions whose areas relate via similarity ratios, yielding a^2 + b^2 = c^2 by equating the areas of the composite figures. This proof emphasizes similarity and minimal dissection for elegance.[28] A proof using differentials adopts an infinitesimal geometric perspective, treating the sides as varying continuously. Consider a right triangle with legs a and b, where the hypotenuse satisfies c = \sqrt{a^2 + b^2}. Differentiating implicitly gives c \, dc = a \, da + b \, db. Multiplying by 2 yields $2c \, dc = 2a \, da + 2b \, db, or d(c^2) = d(a^2) + d(b^2), reflecting small changes along the legs and hypotenuse. This relation confirms the finite form a^2 + b^2 = c^2, bridging geometry with early calculus concepts like the circle equation x^2 + y^2 = c^2. This approach, formalized by John Molokach, underscores the theorem's compatibility with continuous variation.[29] Geometrically, the theorem also emerges from vector relations, where the magnitude of the cross product of two perpendicular leg vectors equals the product of their lengths, tying into the area of the parallelogram they span, which aligns with the squared terms in the theorem. These proofs collectively bridge pure geometry and analytical methods, offering insights into the theorem's robustness across transformations and limits without relying on algebraic expansion.[18]Converse and Related Theorems
Converse of the Theorem
The converse of the Pythagorean theorem states that if a triangle has sides of lengths a, b, and c (with c the longest side) satisfying a^2 + b^2 = c^2, then the angle opposite the side of length c is a right angle.[30] One proof uses the law of cosines, which relates the sides and angles of any triangle: c^2 = a^2 + b^2 - 2ab \cos C, where C is the angle opposite side c. Substituting the given condition a^2 + b^2 = c^2 yields \cos C = 0, implying C = 90^\circ.[31] A geometric proof invokes Thales' theorem, which asserts that an angle inscribed in a semicircle is a right angle. To apply it, construct a circle with diameter equal to the side of length c. Place the endpoints of c at the circle's diameter ends and the third vertex on the circumference. The side lengths satisfy the theorem's condition if and only if the vertex lies on this circle, making the inscribed angle opposite c a right angle by Thales' theorem.[32] Euclid provided an early geometric proof in his Elements (Book I, Proposition 48), constructing an auxiliary right triangle and using congruence to show the original angle must be right.[30] This converse serves as the logical inverse of the original theorem, enabling classification of triangles as right-angled based solely on side lengths.[30] It finds practical use in identifying right triangles from measured dimensions, such as in surveying or engineering.[33]Pythagorean theorem in inner product spaces
The Pythagorean theorem generalizes to the setting of inner product spaces, stating that for any two orthogonal vectors \mathbf{u} and \mathbf{v} in an inner product space, the square of the norm of their sum equals the sum of the squares of their individual norms:\|\mathbf{u} + \mathbf{v}\|^2 = \|\mathbf{u}\|^2 + \|\mathbf{v}\|^2.
This holds in any inner product space, including Euclidean spaces, Hilbert spaces, and other abstract vector spaces equipped with an inner product that induces a norm.[34] The proof relies on the definition of the norm in terms of the inner product. Expanding the left side gives
\|\mathbf{u} + \mathbf{v}\|^2 = \langle \mathbf{u} + \mathbf{v}, \mathbf{u} + \mathbf{v} \rangle = \langle \mathbf{u}, \mathbf{u} \rangle + \langle \mathbf{u}, \mathbf{v} \rangle + \langle \mathbf{v}, \mathbf{u} \rangle + \langle \mathbf{v}, \mathbf{v} \rangle.
Orthogonality implies \langle \mathbf{u}, \mathbf{v} \rangle = 0, and in real inner product spaces, \langle \mathbf{v}, \mathbf{u} \rangle = \langle \mathbf{u}, \mathbf{v} \rangle, so the cross terms vanish, yielding \|\mathbf{u}\|^2 + \|\mathbf{v}\|^2. In complex spaces, the inner product is Hermitian, ensuring the real part of the cross terms is zero under orthogonality.[34] Geometrically, this ties back to the original theorem by interpreting the legs of a right triangle as orthogonal vectors \mathbf{u} and \mathbf{v} in the plane, with the hypotenuse corresponding to their vector sum \mathbf{u} + \mathbf{v}; the norms then represent the side lengths, recovering the relation a^2 + b^2 = c^2.[34] Unlike the converse of the Pythagorean theorem, which applies specifically to triangles in Euclidean geometry to determine right angles from side lengths, this form extends orthogonality and norm relations to arbitrary inner product spaces, independent of dimension or geometric embedding.[34]