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The useful minimum bounding circle of three points is defined either by the circumcircle (where three points are on the minimum bounding circle) or by the two points of the longest side of the triangle (where the two points define a diameter of the circle). It is common to confuse the minimum bounding circle with the circumcircle.
More generally for the same process in d dimensions the average number of neighbors is a constant depending only on d. [5] In the plane, the Delaunay triangulation maximizes the minimum angle. Compared to any other triangulation of the points, the smallest angle in the Delaunay triangulation is at least as large as the smallest angle in any other.
If the parameters l, m, n respectively equal the side lengths a, b, c (or the sines of the angles opposite them) then the equation gives the circumcircle. [1]: p. 199 Each distinct circumconic has a center unique to itself. The equation in trilinear coordinates of the circumconic with center x' : y' : z' is [1]: p. 203
Two sides and an angle not included between them (SSA), if the side length adjacent to the angle is shorter than the other side length. A side and the two angles adjacent to it (ASA) A side, the angle opposite to it and an angle adjacent to it (AAS). For all cases in the plane, at least one of the side lengths must be specified. If only the ...
Case 1: three sides given (SSS). The cosine rule may be used to give the angles A, B, and C but, to avoid ambiguities, the half angle formulae are preferred. Case 2: two sides and an included angle given (SAS). The cosine rule gives a and then we are back to Case 1. Case 3: two sides and an opposite angle given (SSA). The sine rule gives C and ...
The sagitta (also known as the versine) is a line segment drawn perpendicular to a chord, between the midpoint of that chord and the arc of the circle. Given the length y of a chord and the length x of the sagitta, the Pythagorean theorem can be used to calculate the radius of the unique circle that will fit around the two lines: = +.
If a cyclic quadrilateral is also orthodiagonal, the distance from the circumcenter to any side equals half the length of the opposite side. [23] In a cyclic orthodiagonal quadrilateral, the distance between the midpoints of the diagonals equals the distance between the circumcenter and the point where the diagonals intersect. [23]
The vertical side of ∆TVU will have length |A| since the line has slope -A/B. ∆PRS and ∆TVU are similar triangles, since they are both right triangles and ∠PSR ≅ ∠TUV since they are corresponding angles of a transversal to the parallel lines PS and UV (both are vertical lines). [6] Corresponding sides of these triangles are in the ...