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In 2015, an anonymous Japanese woman using the pen name "aerile re" published the first known method (the method of 3 circumcenters) to construct a proof in elementary geometry for a special class of adventitious quadrangles problem. [7] [8] [9] This work solves the first of the three unsolved problems listed by Rigby in his 1978 paper. [5]
X(54) is the Kosnita point of the triangle ABC. In Euclidean geometry, Kosnita's theorem is a property of certain circles associated with an arbitrary triangle.. Let be an arbitrary triangle, its circumcenter and ,, are the circumcenters of three triangles , , and respectively.
Ceva's theorem, case 1: the three lines are concurrent at a point O inside ABC Ceva's theorem, case 2: the three lines are concurrent at a point O outside ABC. In Euclidean geometry, Ceva's theorem is a theorem about triangles.
A polygon is a generalization of a 3-sided triangle, a 4-sided quadrilateral, and so on to n sides. A hypercube is a generalization of a 2-dimensional square, a 3-dimensional cube, and so on to n dimensions. A quadric, such as a hypersphere, ellipsoid, paraboloid, or hyperboloid, is a generalization of a conic section to higher dimensions.
Second generalization: Let a conic S and a point P on the plane. Construct three lines d a , d b , d c through P such that they meet the conic at A, A'; B, B' ; C, C' respectively. Let D be a point on the polar of point P with respect to (S) or D lies on the conic (S).
Ordinary trigonometry studies triangles in the Euclidean plane .There are a number of ways of defining the ordinary Euclidean geometric trigonometric functions on real numbers, for example right-angled triangle definitions, unit circle definitions, series definitions [broken anchor], definitions via differential equations [broken anchor], and definitions using functional equations.
In 2000, Bernard Gibert proposed a generalization of the Lester Theorem involving the Kiepert hyperbola of a triangle. His result can be stated as follows: Every circle with a diameter that is a chord of the Kiepert hyperbola and perpendicular to the triangle's Euler line passes through the Fermat points.
This formula generalizes Heron's formula for the area of a triangle. A triangle may be regarded as a quadrilateral with one side of length zero. From this perspective, as d approaches zero, a cyclic quadrilateral converges into a cyclic triangle (all triangles are cyclic), and Brahmagupta's formula simplifies to Heron's formula.
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