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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.
These rational numbers are the tangents of the individual quarter angles, using the formula for the tangent of the difference of angles. Rational side lengths for the polygon circumscribed by the unit circle are thus obtained as s k = 4q k / (1 + q k 2). The rational area is A = ∑ k 2q k (1 − q k 2) / (1 + q k 2) 2. These can be made into ...
This definition ensures that triangle centers of similar triangles meet the invariance criteria specified above. By convention only the first of the three trilinear coordinates of a triangle center is quoted since the other two are obtained by cyclic permutation of a, b, c. This process is known as cyclicity. [4] [5]
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.
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]
Two triangles are said to be poristic triangles if they have the same incircle and circumcircle. Given a circle with Center O and radius R and another circle with center I and radius r, there are an infinite number of triangles ABC with Circle O(R) as circumcircle and I(r) as incircle if and only if OI 2 = R 2 − 2Rr. These triangles form a ...
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]