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The area of a rational-sided right triangle is called a congruent number, so no congruent number can be square. A right triangle and a square with equal areas cannot have all sides commensurate with each other. There do not exist two integer-sided right triangles in which the two legs of one triangle are the leg and hypotenuse of the other ...
The pons asinorum in Oliver Byrne's edition of the Elements [1]. In geometry, the theorem that the angles opposite the equal sides of an isosceles triangle are themselves equal is known as the pons asinorum (/ ˈ p ɒ n z ˌ æ s ɪ ˈ n ɔːr ə m / PONZ ass-ih-NOR-əm), Latin for "bridge of asses", or more descriptively as the isosceles triangle theorem.
It was also known to be one example of a general rule that any triangle where the length of two sides, each squared and then added together (3 2 + 4 2 = 9 + 16 = 25), equals the square of the length of the third side (5 2 = 25), would also be a right angle triangle.
Specifying two sides and an adjacent angle (SSA), however, can yield two distinct possible triangles unless the angle specified is a right angle. Triangles are congruent if they have all three sides equal (SSS), two sides and the angle between them equal (SAS), or two angles and a side equal (ASA) (Book I, propositions 4, 8, and 26).
For four given elements there is one non-trivial case, which is discussed below. For three given elements there are six cases: three sides, two sides and an included or opposite angle, two angles and an included or opposite side, or three angles. (The last case has no analogue in planar trigonometry.) No single method solves all cases.
The general spherical triangle is fully determined by three of its six characteristics (3 sides and 3 angles). The lengths of the sides a, b, c of a spherical triangle are their central angles, measured in angular units rather than linear units. (On a unit sphere, the angle (in radians) and length around the sphere are numerically the same. On ...
In trigonometry, the law of cotangents is a relationship among the lengths of the sides of a triangle and the cotangents of the halves of the three angles. [1] [2]Just as three quantities whose equality is expressed by the law of sines are equal to the diameter of the circumscribed circle of the triangle (or to its reciprocal, depending on how the law is expressed), so also the law of ...
An automedian triangle is one whose medians are in the same proportions (in the opposite order) as the sides. If x, y, and z are the three sides of a right triangle, sorted in increasing order by size, and if 2x < z, then z, x + y, and y − x are the three sides of an automedian triangle. For instance, the right triangle with side lengths 5 ...
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