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Given three mutually tangent circles, there exist exactly two nonintersecting circles which are tangent circles to all three original circles. These are called the inner and ...

A the (first, or internal) Kenmotu point, also called the congruent squares point, is the triangle center constructed by inscribing three equal squares such that each square ...

Consider the excircles J_A, J_B, and J_C of a triangle, and the external Apollonius circle Gamma tangent externally to all three. Denote the contact point of Gamma and J_A by ...

The first Napoleon point N, also called the outer Napoleon point, is the concurrence of lines drawn between vertices of a given triangle DeltaABC and the opposite vertices of ...

The intersection Fl of the Gergonne line and the Soddy line. In the above figure, D^', E^', and F^' are the Nobbs points, I is the incenter, Ge is the Gergonne point, and S ...

The upper central series of a group G is the sequence of groups (each term normal in the term following it) 1=Z_0<=Z_1<=Z_2<=...<=Z_n<=... that is constructed in the ...

Given triangle DeltaABC, there are four lines simultaneously tangent to the incircle (with center I) and the A-excircle (with center J_A). Of these, three correspond to the ...

Let L, M, and N be lines through A, B, C, respectively, parallel to the Euler line. Let L^' be the reflection of L in sideline BC, let M^' be the reflection of M in sideline ...

The perpendicular bisector of a line segment is the locus of all points that are equidistant from its endpoints. This theorem can be applied to determine the center of a ...

The second Napoleon point N^', also called the inner Napoleon point, is the concurrence of lines drawn between polygon vertices of a given triangle DeltaABC and the opposite ...