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In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle.It states that 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 on the other two sides.
Pythagorean theorem: It states that 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 on the other two sides. The theorem can be written as an equation relating the lengths of the sides a, b and the hypotenuse c, sometimes called the Pythagorean equation: [6]
Angles whose sum is a right angle are called complementary. Complementary angles are formed when a ray shares the same vertex and is pointed in a direction that is in between the two original rays that form the right angle. The number of rays in between the two original rays is infinite. Angles whose sum is a straight angle are supplementary ...
The figure at the right shows three examples beginning with clear inequality (top) and approaching equality (bottom). In the Euclidean case, equality occurs only if the triangle has a 180° angle and two 0° angles, making the three vertices collinear, as shown in the bottom example. Thus, in Euclidean geometry, the shortest distance between ...
Descartes's theorem (plane geometry) Dinostratus' theorem (geometry, analysis) Equal incircles theorem (Euclidean geometry) Euler's quadrilateral theorem ; Euler's theorem in geometry (triangle geometry) Exterior angle theorem (triangle geometry) Feuerbach's theorem ; Finsler–Hadwiger theorem ; Five circles theorem
The law of cosines generalizes the Pythagorean theorem, which holds only for right triangles: if is a right angle then = , and the law of cosines reduces to = + . The law of cosines is useful for solving a triangle when all three sides or two sides and their included angle are given.
A right triangle ABC with its right angle at C, hypotenuse c, and legs a and b,. A right triangle or right-angled triangle, sometimes called an orthogonal triangle or rectangular triangle, is a triangle in which two sides are perpendicular, forming a right angle (1 ⁄ 4 turn or 90 degrees).
Projective geometry is less restrictive than either Euclidean geometry or affine geometry. It is an intrinsically non-metrical geometry, meaning that facts are independent of any metric structure. Under the projective transformations, the incidence structure and the relation of projective harmonic conjugates are preserved.