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Theorem Of The Gnomon
The theorem of the gnomon states that certain parallelograms occurring in a gnomon have areas of equal size. Theorem In a parallelogram ABCD with a point P on the diagonal AC, the parallel to AD through P intersects the side CD in G and the side AB in H. Similarly the parallel to the side AB through P intersects the side AD in I and the side BC in F. Then the theorem of the gnomon states that the parallelograms HBFP and IPGD have equal areas.Lorenz Halbeisen, Norbert Hungerbühler, Juan Läuchli: ''Mit harmonischen Verhältnissen zu Kegelschnitten: Perlen der klassischen Geometrie''. Springer 2016, , pp. 190-191William J. Hazard: ''Generalizations of the Theorem of Pythagoras and Euclid's Theorem of the Gnomon''. The American Mathematical Monthly, volume 36, no. 1 (Jan., 1929), pp. 32–34JSTOR ''Gnomon'' is the name for the L-shaped figure consisting of the two overlapping parallelograms ABFI and AHGD. The parallelograms of equal area HBFP and IPGD are called ''complements'' (o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gnomon Theorem
A gnomon (; ) is the part of a sundial that casts a shadow. The term is used for a variety of purposes in mathematics and other fields. History A painted stick dating from 2300 BC that was excavated at the astronomical site of Taosi is the oldest gnomon known in China. The gnomon was widely used in ancient China from the second century BC onward in order to determine the changes in seasons, orientation, and geographical latitude. The ancient Chinese used shadow measurements for creating calendars that are mentioned in several ancient texts. According to the collection of Zhou Chinese poetic anthologies ''Classic of Poetry'', one of the distant ancestors of King Wen of the Zhou dynasty used to measure gnomon shadow lengths to determine the orientation around the 14th century BC. The ancient Greek philosopher Anaximander (610–546 BC) is credited with introducing this Babylonian instrument to the Ancient Greeks. The ancient Greek mathematician and astronomer Oenopides used th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parallelogram
In Euclidean geometry, a parallelogram is a simple (non- self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure. The congruence of opposite sides and opposite angles is a direct consequence of the Euclidean parallel postulate and neither condition can be proven without appealing to the Euclidean parallel postulate or one of its equivalent formulations. By comparison, a quadrilateral with just one pair of parallel sides is a trapezoid in American English or a trapezium in British English. The three-dimensional counterpart of a parallelogram is a parallelepiped. The etymology (in Greek παραλληλ-όγραμμον, ''parallēl-ógrammon'', a shape "of parallel lines") reflects the definition. Special cases *Rectangle – A parallelogram with four angles of equal size (right angles). *Rhombus – A parallelogram with four sides of eq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gnomon (figure)
In geometry, a gnomon is a plane figure formed by removing a similar parallelogram from a corner of a larger parallelogram; or, more generally, a figure that, added to a given figure, makes a larger figure of the same shape. Building figurate numbers Figurate numbers were a concern of Pythagorean mathematics, and Pythagoras is credited with the notion that these numbers are generated from a ''gnomon'' or basic unit. The gnomon is the piece which needs to be added to a figurate number to transform it to the next bigger one. For example, the gnomon of the square number is the odd number, of the general form 2''n'' + 1, ''n'' = 1, 2, 3, ... . The square of size 8 composed of gnomons looks like this: ~~~~~~~~\begin 8&8&8&8&8&8&8&8\\ 8&7&7&7&7&7&7&7\\ 8&7&6&6&6&6&6&6\\ 8&7&6&5&5&5&5&5\\ 8&7&6&5&4&4&4&4\\ 8&7&6&5&4&3&3&3\\ 8&7&6&5&4&3&2&2\\ 8&7&6&5&4&3&2&1 \end To transform from the ''n-square'' (the square of size ''n'') to the (''n'' + 1)-square, one adjoins 2''n'' + 1 elements: on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Area
Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape A shape or figure is a graphics, graphical representation of an object or its external boundary, outline, or external Surface (mathematics), surface, as opposed to other properties such as color, Surface texture, texture, or material type. A pl ... or planar lamina, while ''surface area'' refers to the area of an open surface or the boundary (mathematics), boundary of a solid geometry, three-dimensional object. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat. It is the two-dimensional analogue of the length of a plane curve, curve (a one-dimensional concept) or the volume of a solid (a three-dimensional concept). The area of a shape can be measured by com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
