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# Geometry Cheatsheet ## Basic Concepts - Geometry is the study of shapes, sizes, positions, and dimensions of objects in space. - Euclidean geometry is the study of geometry based on Euclid's axioms. - Non-Euclidean geometry is the study of geometry that does not satisfy Euclid's axioms. ## Euclidean Geometry - Euclidean geometry is based on five axioms, including the parallel postulate, which states that given a line and a point not on the line, there is exactly one line through the point that is parallel to the given line. - Euclidean geometry includes concepts such as points, lines, planes, angles, circles, and polygons. - The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. ## Non-Euclidean Geometry - Non-Euclidean geometry includes hyperbolic geometry and elliptic geometry. - Hyperbolic geometry satisfies all of Euclid's axioms except the parallel postulate. - Elliptic geometry satisfies all of Euclid's axioms, but the parallel postulate is replaced by its negation. ## Coordinate Geometry - Coordinate geometry is the study of geometry using coordinates. - The distance formula gives the distance between two points in a coordinate system. - The midpoint formula gives the coordinates of the midpoint of a line segment. ## Trigonometry - Trigonometry is the study of the relationships between the sides and angles of triangles. - The sine, cosine, and tangent functions can be used to calculate the lengths of sides and measures of angles in right triangles. - The law of sines and the law of cosines can be used to solve triangles that are not right triangles. ## Transformations - Transformations are functions that map points in a plane to other points in the same plane. - Common transformations include translations, reflections, rotations, and dilations. - Transformations can be used to study symmetry and congruence. ## Resources - [Geometry on Wikipedia](https://en.wikipedia.org/wiki/Geometry) - [Euclid's Elements](https://mathcs.clarku.edu/~djoyce/java/elements/toc.html) (online version) - [Introduction to Geometry by H. S. M. Coxeter](https://www.wiley.com/en-us/Introduction+to+Geometry%2C+2nd+Edition-p-9780471504580)