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The direct theorem was Proposition 22 in Book 3 of Euclid's Elements. [3] Equivalently, a convex quadrilateral is cyclic if and only if each exterior angle is equal to the opposite interior angle . In 1836 Duncan Gregory generalized this result as follows: Given any convex cyclic 2 n -gon, then the two sums of alternate interior angles are each ...
In geometry, the circumscribed circle or circumcircle of a triangle is a circle that passes through all three vertices.The center of this circle is called the circumcenter of the triangle, and its radius is called the circumradius.
In geometry, a triangle center or triangle centre is a point in the triangle's plane that is in some sense in the middle of the triangle. For example, the centroid, circumcenter, incenter and orthocenter were familiar to the ancient Greeks, and can be obtained by simple constructions.
In trigonometry, the law of sines, sine law, sine formula, or sine rule is an equation relating the lengths of the sides of any triangle to the sines of its angles. According to the law, = = =, where a, b, and c are the lengths of the sides of a triangle, and α, β, and γ are the opposite angles (see figure 2), while R is the radius of the triangle's circumcircle.
The nine-point center is the circumcenter of the medial triangle of the given triangle, the circumcenter of the orthic triangle of the given triangle, and the circumcenter of the Euler triangle. More generally it is the circumcenter of any triangle defined from three of the nine points defining the nine-point circle. [citation needed]
As of 16 October 2024, the list identifies over 65,000 triangle centers [2] and is managed cooperatively by an international team of geometry researchers. [3] This resource is seen as a pillar of the modern geometry. [4] In GeoGebra, this encyclopedia is provided at fingertip by a special command. [5]
The first four partial sums of the series 1 + 2 + 3 + 4 + ⋯.The parabola is their smoothed asymptote; its y-intercept is −1/12. [1]The infinite series whose terms ...
Likewise, a triangle's circumcenter—the intersection of the three sides' perpendicular bisectors, which is the center of the circle that passes through all three vertices—falls inside an acute triangle but outside an obtuse triangle. The right triangle is the in-between case: both its circumcenter and its orthocenter lie on its boundary.