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The radius of an excircle. Let a triangle have exradius r_A (sometimes denoted rho_A), opposite side of length a and angle A, area Delta, and semiperimeter s. Then r_1 = ...

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 extangents circle is the circumcircle of the extangents triangle. Its center function is a complicated 9th-order polynomial and its circle function is a complicated ...

Let J_A, J_B, and J_C be the vertices of the outer Soddy triangle, and also let E_A, E_B, and E_C be the pairwise contact points of the three tangent circles. Then the lines ...

Consider Kimberling centers X_(20) (de Longchamps point Z; intersection L_S intersection L_E of the Soddy line and Euler line), X_(468) (intersection L_E intersection L_O of ...

Gauss's theorema egregium states that the Gaussian curvature of a surface embedded in three-space may be understood intrinsically to that surface. "Residents" of the surface ...

The pedal curve of a rectangular hyperbola with the pedal point at the focus is a circle (left figure; Hilbert and Cohn-Vossen 1999, p. 26). The pedal curve of a rectangular ...

Given a triangle DeltaA_1A_2A_3, the points A_1, I, and J_1 lie on a line, where I is the incenter and J_1 is the excenter corresponding to A_1. Furthermore, the circle with ...

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 ...

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 ...