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It is easy to find an n-equidissection of a triangle for all n.As a result, if a polygon has an m-equidissection, then it also has an mn-equidissection for all n.In fact, often a polygon's spectrum consists precisely of the multiples of some number m; in this case, both the spectrum and the polygon are called principal and the spectrum is denoted . [2]
A partition into triangles of equal area is called an equidissection. Most polygons cannot be equidissected, and those that can often have restrictions on the possible numbers of triangles. For example, Monsky's theorem states that there is no odd equidissection of a square. [1]
The area of a triangle can be demonstrated, for example by means of the congruence of triangles, as half of the area of a parallelogram that has the same base length and height. A graphic derivation of the formula T = h 2 b {\displaystyle T={\frac {h}{2}}b} that avoids the usual procedure of doubling the area of the triangle and then halving it.
The most common version uses the concept of "equidecomposability" of polygons: two polygons are equidecomposable if they can be split into finitely many triangles that only differ by some isometry (in fact only by a combination of a translation and a rotation). In this case the Wallace–Bolyai–Gerwien theorem states that two polygons are ...
the inverse geodesic problem or second geodesic problem, given A and B, determine s 12, α 1, and α 2. As can be seen from Fig. 1, these problems involve solving the triangle NAB given one angle, α 1 for the direct problem and λ 12 = λ 2 − λ 1 for the inverse problem, and its two adjacent sides.
All problems that can be solved using mass point geometry can also be solved using either similar triangles, vectors, or area ratios, [2] but many students prefer to use mass points. Though modern mass point geometry was developed in the 1960s by New York high school students, [ 3 ] the concept has been found to have been used as early as 1827 ...
Because each special triangle has area , a polygon of area will be subdivided into special triangles. [ 5 ] The subdivision of the polygon into triangles forms a planar graph , and Euler's formula V − E + F = 2 {\displaystyle V-E+F=2} gives an equation that applies to the number of vertices, edges, and faces of any planar graph.
Solution of triangles (Latin: solutio triangulorum) is the main trigonometric problem of finding the characteristics of a triangle (angles and lengths of sides), when some of these are known. The triangle can be located on a plane or on a sphere. Applications requiring triangle solutions include geodesy, astronomy, construction, and navigation.