"four elementary lemmata: Next, each top square is related to a triangle congruent with another triangle related in turn to one of two rectangles making up the lower square. The proof is as follows: This proof, which appears in Euclid's ""Elements"" as that of Proposition 47 in Book 1, demonstrates that the area of the square on the hypotenuse is the sum of the areas of the other two squares. This is quite distinct from the proof by similarity of triangles, which is conjectured to be the proof that Pythagoras used. We have already discussed the Pythagorean proof, which was"
"proposition 47) states that in any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle). Thales' theorem, named after Thales of Miletus states that if A, B, and C are points on a circle where the line AC is a diameter of the circle, then the angle ABC is a right angle. Cantor supposed that Thales proved his theorem by means of Euclid Book I,"
"altitude to the hypotenuse is related to the legs of the right triangle by For solutions of this equation in integer values of ""a, b, f"", and ""c"", see here. The altitude from either leg coincides with the other leg. Since these intersect at the right-angled vertex, the right triangle's orthocenter—the intersection of its three altitudes—coincides with the right-angled vertex. The Pythagorean theorem states that: In any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the"
"the hypotenuse equals the sum of the squares of the lengths of the two other sides. If the hypotenuse has length ""c"", and the legs have lengths ""a"" and ""b"", then the theorem states that The converse is true: if the lengths of the sides of a triangle satisfy the above equation, then the triangle has a right angle opposite side ""c"". Some other facts about right triangles: For all triangles, angles and sides are related by the law of cosines and law of sines (also called the ""cosine rule"" and ""sine rule""). The triangle inequality states that the sum"
"to a square. In either case, it states that the square of the hypotenuse equals the sum of the squares of the sides. If restricted to right-angled isosceles triangles, however, it would constitute a less general claim, but the text seems to be quite open to unequal sides. If this refers to a rectangle, it is the earliest recorded statement of the Pythagorean theorem. Baudhāyana also provides a non-axiomatic demonstration using a rope measure of the reduced form of the Pythagorean theorem for an isosceles right triangle: Sequences of Pythagorean triples used in cryptography as random sequences and for the"
"and the remaining square. Putting the two rectangles together to reform the square on the hypotenuse, its area is the same as the sum of the area of the other two squares. The details follow. Let ""A"", ""B"", ""C"" be the vertices of a right triangle, with a right angle at ""A"". Drop a perpendicular from ""A"" to the side opposite the hypotenuse in the square on the hypotenuse. That line divides the square on the hypotenuse into two rectangles, each having the same area as one of the two squares on the legs. For the formal proof, we require"
"opposite the right angle corner is the sum of the squares of the areas of the other three faces. This result can be generalized as in the """"n""-dimensional Pythagorean theorem"": This statement is illustrated in three dimensions by the tetrahedron in the figure. The ""hypotenuse"" is the base of the tetrahedron at the back of the figure, and the ""legs"" are the three sides emanating from the vertex in the foreground. As the depth of the base from the vertex increases, the area of the ""legs"" increases, while that of the base is fixed. The theorem suggests that when this"
"triangle to the square of its hypotenuse is the same for similar triangles, the relationship between the areas of the three triangles holds for the squares of the sides of the large triangle as well. The theorem can be proved algebraically using four copies of a right triangle with sides ""a"", ""b"" and ""c"", arranged inside a square with side ""c"" as in the top half of the diagram. The triangles are similar with area formula_10, while the small square has side and area . The area of the large square is therefore But this is a square with side"
"Hypotenuse In geometry, a hypotenuse is the longest side of a right-angled triangle, the side opposite of the right angle. The length of the hypotenuse of a right triangle can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse equals the sum of the squares of the lengths of the other two sides. For example, if one of the other sides has a length of 3 (when squared, 9) and the other has a length of 4 (when squared, 16), then their squares add up to 25. The length of the hypotenuse"
"Geometric mean theorem The right triangle altitude theorem or geometric mean theorem is a result in elementary geometry that describes a relation between the lengths of the altitude on the hypotenuse in a right triangle and the two line segments it creates on the hypotenuse. It states that the geometric mean of the two segments equals the altitude. If ""h"" denotes the altitude in a right triangle and ""p"" and ""q"" the segments on the hypotenuse then the theorem can be stated as: or in term of areas: The latter version yields a method to square a rectangle with ruler"
"be made quickly and without technical instruments. The geometric law behind the measurement is the Pythagorean theorem (""The square of the hypotenuse of a right triangle is equal to the sum of the squares of the two adjacent sides""). Thales' theorem states that an angle inscribed in a semicircle (with a vertex on the semicircle and its defining rays going through the endpoints of the semicircle) is a right angle. Two application examples in which the right angle and the Thales' theorem are included (see animations). Right angle In geometry and trigonometry, a right angle is an angle of exactly"
"formula_9 One may also obtain the value of the angle formula_9 by the equation: in which formula_16 is the other cathetus. Hypotenuse In geometry, a hypotenuse is the longest side of a right-angled triangle, the side opposite of the right angle. The length of the hypotenuse of a right triangle can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse equals the sum of the squares of the lengths of the other two sides. For example, if one of the other sides has a length of 3 (when squared, 9) and the"
"to complete it, the other a rectangle of area ""pq"". Since both arrangements yield the same triangle, the areas of the square and the rectangle must be identical. The square of the altitude can be transformed into an rectangle of equal area with sides ""p"" and ""q"" with the help of three shear mappings (shear mappings preserve the area): Geometric mean theorem The right triangle altitude theorem or geometric mean theorem is a result in elementary geometry that describes a relation between the lengths of the altitude on the hypotenuse in a right triangle and the two line segments it"
"circle, then the area of the square on the side of the pentagon equals the sum of the areas of the squares on the sides of the hexagon and the decagon inscribed in the same circle. He used Ptolemy's theorem on quadrilaterals inscribed in a circle to derive formulas for the chord of a half-arc, the chord of the sum of two arcs, and the chord of a difference of two arcs. The theorem states that for a quadrilateral inscribed in a circle, the product of the lengths of the diagonals equals the sum of the products of the two"
"says which is de Gua's theorem. Jean Paul de Gua de Malves (1713–85) published the theorem in 1783, but around the same time a slightly more general version was published by another French mathematician, Charles de Tinseau d'Amondans (1746–1818), as well. However the theorem had also been known much earlier to Johann Faulhaber (1580–1635) and René Descartes (1596–1650). De Gua's theorem De Gua's theorem is a three-dimensional analog of the Pythagorean theorem and named after Jean Paul de Gua de Malves. If a tetrahedron has a right-angle corner (like the corner of a cube), then the square of the area"
"De Gua's theorem De Gua's theorem is a three-dimensional analog of the Pythagorean theorem and named after Jean Paul de Gua de Malves. If a tetrahedron has a right-angle corner (like the corner of a cube), then the square of the area of the face opposite the right-angle corner is the sum of the squares of the areas of the other three faces. The Pythagorean theorem and de Gua's theorem are special cases (""n"" = 2, 3) of a general theorem about ""n""-simplices with a right-angle corner. This, in turn, is a special case of a yet more general theorem"
"the lengths of any two sides are known the length of the third side can be found. Another corollary of the theorem is that in any right triangle, the hypotenuse is greater than any one of the other sides, but less than their sum. A generalization of this theorem is the law of cosines, which allows the computation of the length of any side of any triangle, given the lengths of the other two sides and the angle between them. If the angle between the other sides is a right angle, the law of cosines reduces to the Pythagorean equation."
"angles, the enclosed angle and the form corresponding to Pythagoras's theorem is regained: formula_34 The Pythagorean theorem, valid for right triangles, therefore is a special case of the more general law of cosines, valid for arbitrary triangles. In a right triangle with sides ""a"", ""b"" and hypotenuse ""c"", trigonometry determines the sine and cosine of the angle ""θ"" between side ""a"" and the hypotenuse as: From that it follows: where the last step applies Pythagoras's theorem. This relation between sine and cosine is sometimes called the fundamental Pythagorean trigonometric identity. In similar triangles, the ratios of the sides are the"
"of two legs instead of the hypotenuse (see #Similar figures on the three sides). In Einstein's proof, the shape that includes the hypotenuse is the right triangle itself. The dissection consists of dropping a perpendicular from the vertex of the right angle of the triangle to the hypotenuse, thus splitting the whole triangle into two parts. Those two parts have the same shape as the original right triangle, and have the legs of the original triangle as their hypotenuses, and the sum of their areas is that of the original triangle. Because the ratio of the area of a right"
"then with equality only in the isosceles case. If segments of lengths ""p"" and ""q"" emanating from vertex ""C"" trisect the hypotenuse into segments of length ""c""/3, then The right triangle is the only triangle having two, rather than one or three, distinct inscribed squares. Let ""h"" and ""k"" (""h"" > ""k"") be the sides of the two inscribed squares in a right triangle with hypotenuse ""c"". Then These sides and the incircle radius ""r"" are related by a similar formula: The perimeter of a right triangle equals the sum of the radii of the incircle and the three excircles:"
"it, subtracts unity and makes half the difference the greater of the sides about the right angle; lastly it adds unity to this and so forms the remaining side, the hypotenuse. ...For the method of Plato argues from even numbers. It takes the given even number and makes it one of the sides about the right angle; then, bisecting this number and squaring the half, it adds unity to the square to form the hypotenuse, and subtracts unity from the square to form the other side about the right angle. ... Thus it has formed the same triangle that which"
"triangles and for them there exists an ""n""-dimensional version of the Pythagorean theorem: The sum of the squared (""n"" − 1)-dimensional volumes of the facets adjacent to the orthogonal corner equals the squared (""n"" − 1)-dimensional volume of the facet opposite of the orthogonal corner. where formula_67 are facets being pairwise orthogonal to each other but not orthogonal to formula_68, which is the facet opposite the orthogonal corner. For a 2-simplex the theorem is the Pythagorean theorem for triangles with a right angle and for a 3-simplex it is de Gua's theorem for a tetrahedron with an orthogonal corner. The"
"Apollonius's theorem In geometry, Apollonius's theorem is a theorem relating the length of a median of a triangle to the lengths of its side. It states that ""the sum of the squares of any two sides of any triangle equals twice the square on half the third side, together with twice the square on the median bisecting the third side"". Specifically, in any triangle , if is a median, then It is a special case of Stewart's theorem. For an isosceles triangle with , the median is perpendicular to and the theorem reduces to the Pythagorean theorem for triangle (or"
"four sides equal to ; (we have a right triangle whose hypotenuse is and the side opposite to the angle is ). The pale lines correspond to the small diagonal of the rhombus. As the diagonals are the bisectors of the neighbouring sides, we can see that the pale line makes an angle equal to with the perpendicular of each pattern's line. Additionally, the spacing between two pale lines is , half of the long diagonal. The long diagonal is the hypotenuse of a right triangle and the sides of the right angle are and . The Pythagorean theorem gives:"
"side ""AB"". Point ""H"" divides the length of the hypotenuse ""c"" into parts ""d"" and ""e"". The new triangle ""ACH"" is similar to triangle ""ABC"", because they both have a right angle (by definition of the altitude), and they share the angle at ""A"", meaning that the third angle will be the same in both triangles as well, marked as ""θ"" in the figure. By a similar reasoning, the triangle ""CBH"" is also similar to ""ABC"". The proof of similarity of the triangles requires the triangle postulate: the sum of the angles in a triangle is two right angles, and"
"the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. The students of Pythagoras were known as pythagoreans. Pythagoreans followed a very specific way of life. They were famous for friendship, unselfishness, and honesty. The pythagoreans also believed in a life after the current which drove them to be people who have no attachment to personal possessions everything was communal; they were also vegetarians. The people in a pythagorean society were known as mathematikoi. There are two forms that Pythagoras taught, Exoteric and Esoteric. Exoteric was the teaching of"
"which up to similarity we have sides formula_3 and area formula_4 Ono's inequality In mathematics, Ono's inequality is a theorem about triangles in the Euclidean plane. In its original form, as conjectured by T. Ono in 1914, the inequality is actually false; however, the statement is true for acute triangles and right triangles, as shown by Balitrand in 1916. Consider an acute or right triangle in the Euclidean plane with side lengths ""a"", ""b"" and ""c"" and area ""A"". Then This inequality fails for general triangles (to which Ono's original conjecture applied), as shown by the counterexample formula_2 The inequality"
"Finsler–Hadwiger theorem The Finsler–Hadwiger theorem is statement in Euclidean plane geometry that describes a third square derived from any two squares that share a vertex. The theorem is named after Paul Finsler and Hugo Hadwiger, who published it in 1937 as part of the same paper in which they published the Hadwiger–Finsler inequality relating the side lengths and area of a triangle. To state the theorem, suppose that ABCD and AB'C'D' are two squares with common vertex A. Let E and G be the midpoints of B'D and D'B respectively, and let F and H be the centers of the"
"angle across to make a right angle with one side, the right angle triangle's hypotenuse is length one and the sides are of length and . From this the trigonometric function tangent of 60° equals , and the sine of 60° and the cosine of 30° both equal . The square root of 3 also appears in algebraic expressions for various other trigonometric constants, including the sines of 3°, 12°, 15°, 21°, 24°, 33°, 39°, 48°, 51°, 57°, 66°, 69°, 75°, 78°, 84°, and 87°. It is the distance between parallel sides of a regular hexagon with sides of length"
"of which matches the area of the quarter circle. This rectangle can be transformed into a square of the same area with the help of Euclid's geometric mean theorem. One extends the side by a line segment formula_13 and draws a half circle to right of , which has as its diameter. The extension of meets the half circle in ""R"" and due to Thales' theorem the line segment is the altitude of the right angle triangle ""QNR"". Hence the geometric mean theorem can be applied, which means that forms the side of a square ""OUSR"" with the same area"
"terms of side length ""a"" can be derived directly using the Pythagorean theorem or using trigonometry. The area of a triangle is half of one side ""a"" times the height ""h"" from that side: The legs of either right triangle formed by an altitude of the equilateral triangle are half of the base ""a"", and the hypotenuse is the side ""a"" of the equilateral triangle. The height of an equilateral triangle can be found using the Pythagorean theorem so that Substituting ""h"" into the area formula (1/2)""ah"" gives the area formula for the equilateral triangle: Using trigonometry, the area of"
"two squares. Then the theorem states that the quadrilateral EFGH is a square as well. Finsler–Hadwiger theorem The Finsler–Hadwiger theorem is statement in Euclidean plane geometry that describes a third square derived from any two squares that share a vertex. The theorem is named after Paul Finsler and Hugo Hadwiger, who published it in 1937 as part of the same paper in which they published the Hadwiger–Finsler inequality relating the side lengths and area of a triangle. To state the theorem, suppose that ABCD and AB'C'D' are two squares with common vertex A. Let E and G be the midpoints"
"any orthodiagonal quadrilateral, the sum of the squares of two opposite sides equals that of the other two opposite sides: for successive sides ""a"", ""b"", ""c"", and ""d"", we have This follows from the Pythagorean theorem, by which either of these two sums of two squares can be expanded to equal the sum of the four squared distances from the quadrilateral's vertices to the point where the diagonals intersect. Conversely, any quadrilateral in which ""a"" + ""c"" = ""b"" + ""d"" must be orthodiagonal. This can be proved in a number of ways, including using the law of cosines, vectors,"