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Dehn's proof is an instance in which abstract algebra is used to prove an impossibility result in geometry.Other examples are doubling the cube and trisecting the angle.. Two polyhedra are called scissors-congruent if the first can be cut into finitely many polyhedral pieces that can be reassembled to yield the second.
The only integer triangle with three rational angles (rational numbers of degrees, or equivalently rational fractions of a full turn) is the equilateral triangle. [2] This is because integer sides imply three rational cosines by the law of cosines , and by Niven's theorem a rational cosine coincides with a rational angle if and only if the ...
A quadrilateral such as BCEF is called an adventitious quadrangle when the angles between its diagonals and sides are all rational angles, angles that give rational numbers when measured in degrees or other units for which the whole circle is a rational number. Numerous adventitious quadrangles beyond the one appearing in Langley's puzzle have ...
If A, B are two points on a line a, and if A′ is a point upon the same or another line a′, then, upon a given side of A′ on the straight line a′, we can always find a point B′ so that the segment AB is congruent to the segment A′B′. We indicate this relation by writing AB ≅ A′B′.
In geometry, a Heronian triangle (or Heron triangle) is a triangle whose side lengths a, b, and c and area A are all positive integers. [ 1 ][ 2 ] Heronian triangles are named after Heron of Alexandria, based on their relation to Heron's formula which Heron demonstrated with the example triangle of sides 13, 14, 15 and area 84. 3.
Congruence permits alteration of some properties, such as location and orientation, but leaves others unchanged, like distances and angles. The unchanged properties are called invariants. In geometry, two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the mirror image of the other.
If a is replaced with the fraction m/n in the sequence, the result is equal to the 'standard' triple generator (2mn, m 2 − n 2, m 2 + n 2) after rescaling. It follows that every triple has a corresponding rational a value which can be used to generate a similar triangle (one with the same three angles and with sides in the same proportions as ...
For a spherical triangle, the sum of the angles is greater than 180° and can be up to 540°. The amount by which the sum of the angles exceeds 180° is called the spherical excess, denoted as Ε or Δ. [7] Specifically, the sum of the angles is 180° × (1 + 4f), where f is the fraction of the sphere's area which is enclosed by the triangle.
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