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Analytic geometry allows the study of curves unrelated to circles and lines. Such curves can be defined as the graph of functions, the study of which led to differential geometry. They can also be defined as implicit equations, often polynomial equations (which spawned algebraic geometry). Analytic geometry also makes it possible to consider ...
In analytic geometry, congruence may be defined intuitively thus: two mappings of figures onto one Cartesian coordinate system are congruent if and only if, for any two points in the first mapping, the Euclidean distance between them is equal to the Euclidean distance between the corresponding points in the second mapping.
In the former case, equivalence of two definitions means that a mathematical object (for example, geometric body) satisfies one definition if and only if it satisfies the other definition. In the latter case, the meaning of equivalence (between two definitions of a structure) is more complicated, since a structure is more abstract than an object.
Secant-, chord-theorem. For the intersecting secants theorem and chord theorem the power of a point plays the role of an invariant: . Intersecting secants theorem: For a point outside a circle and the intersection points , of a secant line with the following statement is true: | | | | = (), hence the product is independent of line .
An example of a circular triangle with three convex edges is a Reuleaux triangle, which can be made by intersecting three circles of equal size. The construction may be performed with a compass alone without needing a straightedge, by the Mohr–Mascheroni theorem. Alternatively, it can be constructed by rounding the sides of an equilateral ...
Antecedent of Playfair's axiom: a line and a point not on the line Consequent of Playfair's axiom: a second line, parallel to the first, passing through the point. In geometry, Playfair's axiom is an axiom that can be used instead of the fifth postulate of Euclid (the parallel postulate):
The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia and Egypt in the 2nd millennium BC. [5] [6] Early geometry was a collection of empirically discovered principles concerning lengths, angles, areas, and volumes, which were developed to meet some practical need in surveying, construction, astronomy, and
If the means of the geometric proportion are equal, and the rightmost extreme is equal to the difference between the leftmost extreme and a mean, then such a proportion is called harmonic: [6]: =: (). In this case the ratio : is called golden ratio.
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