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A group set is a set whose elements are acted on by a group. If the group G acts on the set S, then S is called a G-set. Let G be a group and let S be a G-set. Then for every ...

Let C be a curve, let O be a fixed point (the pole), and let O^' be a second fixed point. Let P and P^' be points on a line through O meeting C at Q such that P^'Q=QP=QO^'. ...

The Kiepert hyperbola is a hyperbola and triangle conic that is related to the solution of Lemoine's problem and its generalization to isosceles triangles constructed on the ...

The Bevan circle, sometimes also called the excentral circle, is the circumcircle of the excentral triangle of the reference triangle, i.e., it is the circle passing through ...

The line segment KO^_ joining the symmedian point K and circumcenter O of a given triangle. It is the diameter of the triangle's Brocard circle, and lies along the Brocard ...

A Brocard line is a line from any of the vertices A_i of a triangle DeltaA_1A_2A_3 to the first Omega or second Omega^' Brocard point. Let the angle at a vertex A_i also be ...

The triangulation point Y of a reference triangle DeltaABC for which triangles DeltaBYC, DeltaCYA, and DeltaAYB have congruent incircles. It is a special case of an Elkies ...

The center J_i of an excircle. There are three excenters for a given triangle, denoted J_1, J_2, J_3. The incenter I and excenters J_i of a triangle are an orthocentric ...

Given triangle DeltaABC, there are four lines simultaneously tangent to the B- and C-excircles (with centers J_B and J_C, respectively). Of these, three correspond to the ...

The inner Napoleon circle, a term coined here for the first time, is the circumcircle of the inner Napoleon triangle. It has center at the triangle centroid G (and is thus ...