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Let P and U be points, neither of which lies on a sideline of DeltaABC, given in barycentric coordinates by P=p:q:r and U=u:v:w. The P-reciprocal conjugate of U is then the ...

Let theta be an irrational number, define S(theta)={c+dtheta:c,d in N}, and let c_n(theta)+thetad_n(theta) be the sequence obtained by arranging the elements of S(theta) in ...

The Simson cubic is the triangle cubic that is the locus of tripoles of the Simson lines of a triangle DeltaABC. It has trilinear equation ...

A line can be specified in trilinear coordinates by parameters (l,m,n) such that the trilinear coordinates alpha:beta:gamma obey lalpha+mbeta+ngamma=0. (1) The trilinear line ...

Bipolar coordinates are a two-dimensional system of coordinates. There are two commonly defined types of bipolar coordinates, the first of which is defined by x = ...

A projection of a figure by parallel rays. In such a projection, tangencies are preserved. Parallel lines project to parallel lines. The ratio of lengths of parallel segments ...

At the points where a line X cuts the sides of a triangle DeltaA_1A_2A_3, draw three perpendiculars to the sides, one through each point of intersection. The resulting three ...

If three conics pass through two given points Q and Q^', then the lines joining the other two intersections of each pair of conics P_(ij)P_(ij)^' are concurrent at a point X ...

The unique (modulo rotations) scalene triangle formed from three vertices of a regular heptagon, having vertex angles pi/7, 2pi/7, and 4pi/7. There are a number of amazing ...

A geometric construction done with a movable compass alone. All constructions possible with a compass and straightedge are possible with a movable compass alone, as was ...