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With a the shorter and b the longer legs of a triangle and c its hypotenuse, the Pythagoras family of triplets is defined by c − b = 1, the Plato family by c − b = 2, and the Fermat family by | a − b | = 1. The Stifel sequence produces all primitive triplets of the Pythagoras family, and the Ozanam sequence produces all primitive triples ...
A sequence of possible hypotenuse numbers for a primitive Pythagorean triple can be found at (sequence A008846 in the OEIS). The area (K = ab/2) is a congruent number [17] divisible by 6. In every Pythagorean triangle, the radius of the incircle and the radii of the three excircles are positive integers.
If any of the above matrices, say A, is applied to a triple (a, b, c) T having the Pythagorean property a 2 + b 2 = c 2 to obtain a new triple (d, e, f) T = A(a, b, c) T, this new triple is also Pythagorean.
The Pythagorean prime 5 and its square root are both hypotenuses of right triangles with integer legs. The formulas show how to transform any right triangle with integer legs into another right triangle with integer legs whose hypotenuse is the square of the first triangle's hypotenuse.
The Pythagorean theorem was known and used by the Babylonians and Indians centuries before Pythagoras, [216] [214] [217] [218] but he may have been the first to introduce it to the Greeks. [219] [217] Some historians of mathematics have even suggested that he—or his students—may have constructed the first proof. [220]
The term sister group is used in phylogenetic analysis, however, only groups identified in the analysis are labeled as "sister groups".. An example is birds, whose commonly cited living sister group is the crocodiles, but that is true only when discussing extant organisms; [3] [4] when other, extinct groups are considered, the relationship between birds and crocodiles appears distant.
For example, in a full tree of life, the entire clade of animals can be collapsed to a single branch of the tree. However, this is merely a limitation of rendering space. In theory, a true and complete tree for all living organisms or for any DNA sequence could be generated. [4] Nevertheless, phylogenies can sometimes appear in a non-treelike form.
The idea of a tree of life arose from ancient notions of a ladder-like progression from lower into higher forms of life (such as in the Great Chain of Being).Early representations of "branching" phylogenetic trees include a "paleontological chart" showing the geological relationships among plants and animals in the book Elementary Geology, by Edward Hitchcock (first edition: 1840).