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Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry, Elements. Euclid's approach consists in assuming a small set of intuitively appealing axioms (postulates) and deducing many other propositions ( theorems ) from these.
In mathematics, the simplest form of the parallelogram law (also called the parallelogram identity) belongs to elementary geometry. It states that the sum of the squares of the lengths of the four sides of a parallelogram equals the sum of the squares of the lengths of the two diagonals. We use these notations for the sides: AB, BC, CD, DA.
In Euclidean geometry, Ceva's theorem is a theorem about triangles. Given a triangle ABC, let the lines AO, BO, CO be drawn from the vertices to a common point O (not on one of the sides of ABC), to meet opposite sides at D, E, F respectively. (The segments AD, BE, CF are known as cevians.) Then, using signed lengths of segments,
The Upper and Lower Dimension axioms together require that any model of these axioms have dimension 2, i.e. that we are axiomatizing the Euclidean plane. Suitable changes in these axioms yield axiom sets for Euclidean geometry for dimensions 0, 1, and greater than 2 (Tarski and Givant 1999: Axioms 8 (1), 8 (n), 9 (0), 9 (1), 9 (n)).
The Euclidean algorithm was probably invented before Euclid, depicted here holding a compass in a painting of about 1474. The Euclidean algorithm is one of the oldest algorithms in common use. [27] It appears in Euclid's Elements (c. 300 BC), specifically in Book 7 (Propositions 1–2) and Book 10 (Propositions 2–3). In Book 7, the algorithm ...
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.
The butterfly theorem is a classical result in Euclidean geometry, which can be stated as follows: [1]: p. 78 Let M be the midpoint of a chord PQ of a circle, through which two other chords AB and CD are drawn; AD and BC intersect chord PQ at X and Y correspondingly. Then M is the midpoint of XY.
The books cover plane and solid Euclidean geometry, elementary number theory, and incommensurable lines. Elements is the oldest extant large-scale deductive treatment of mathematics. It has proven instrumental in the development of logic and modern science , and its logical rigor was not surpassed until the 19th century.
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