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The circumcircle of the Fuhrmann triangle. It has the line HNa, where H is the orthocenter and Na is the Nagel point, as its diameter. In fact, these points (Kimberling ...

The circle H which touches the incircles I, I_A, I_B, and I_C of a circular triangle ABC and its associated triangles. It is either externally tangent to I and internally ...

Eliminate each knot crossing by connecting each of the strands coming into the crossing to the adjacent strand leaving the crossing. The resulting strands no longer cross but ...

The Brocard circle, also known as the seven-point circle, is the circle having the line segment connecting the circumcenter O and symmedian point K of a triangle DeltaABC as ...

Draw antiparallels through the symmedian point K. The points where these lines intersect the sides then lie on a circle, known as the cosine circle (or sometimes the second ...

There are four completely different definitions of the so-called Apollonius circles: 1. The set of all points whose distances from two fixed points are in a constant ratio ...

The points of tangency t_1 and t_2 for the four lines tangent to two circles with centers x_1 and x_2 and radii r_1 and r_2 are given by solving the simultaneous equations ...

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 common incircle of the medial triangle DeltaM_AM_BM_C (left figure) and the congruent triangle DeltaQ_AQ_BQ_C, where Q_i are the midpoints of the line segment joining the ...

The locus of the similitude center of two circles.