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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.
Circular segment - the part of the sector that remains after removing the triangle formed by the center of the circle and the two endpoints of the circular arc on the boundary. Scale of chords; Ptolemy's table of chords; Holditch's theorem, for a chord rotating in a convex closed curve; Circle graph; Exsecant and excosecant
Circular triangles give the solution to an isoperimetric problem in which one seeks a curve of minimum length that encloses three given points and has a prescribed area. . When the area is at least as large as the circumcircle of the points, the solution is any circle of that area surrounding the poi
The subset of the Reuleaux triangle consisting of points belonging to three or more diameters is the interior of the larger of these two triangles; it has a larger area than the set of three-diameter points of any other curve of constant width. [16] Centrally symmetric shapes inside and outside a Reuleaux triangle, used to measure its asymmetry
If the interior point is the circumcenter of the reference triangle, the vertices of the pedal triangle are the midpoints of the reference triangle's sides, and so the pedal triangle is called the midpoint triangle or medial triangle. The midpoint triangle subdivides the reference triangle into four congruent triangles which are similar to the ...
Then any point P associated with the reference triangle ABC can be defined in a Cartesian system as a vector = +. If this point P has trilinear coordinates x : y : z then the conversion formula from the coefficients k 1 and k 2 in the Cartesian representation to the trilinear coordinates is, for side lengths a, b, c opposite vertices A, B, C ,
The points (α, β) are plotted as with Newton's diagram method but the line α+β=n, where n is the degree of the curve, is added to form a triangle which contains the diagram. This method considers all lines which bound the smallest convex polygon which contains the plotted points (see convex hull ).
If the three points are collinear, R can be informally considered to be +∞, and it makes rigorous sense to define c(x, y, z) = 0. If any of the points x, y and z are coincident, again define c(x, y, z) = 0. Using the well-known formula relating the side lengths of a triangle to its area, it follows that