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The Lemoine cubic is the triangle cubic with trilinear equation It passes through Kimberling centers X_n for n=3, 4, 32, 56, and 1147.

For a triangle DeltaABC and three points A^', B^', and C^', one on each of its sides, the three Miquel circles are the circles passing through each polygon vertex and its ...

Let a line in three dimensions be specified by two points x_1=(x_1,y_1,z_1) and x_2=(x_2,y_2,z_2) lying on it, so a vector along the line is given by v=[x_1+(x_2-x_1)t; ...

The Pythagoras tree is a fractal constructed iteratively from a right triangle with squares erected on each of the sides. Subsequent iterations add additional similar right ...

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 ...

A rotor is a convex figure that can be rotated inside a polygon (or polyhedron) while always touching every side (or face). The least area rotor in a square is the Reuleaux ...

The semiperimeter on a figure is defined as s=1/2p, (1) where p is the perimeter. The semiperimeter of polygons appears in unexpected ways in the computation of their areas. ...

The difference between the sum of the angles A, B, and C of a spherical triangle and pi radians (180 degrees), E=A+B+C-pi. The notation Delta is sometimes used for spherical ...

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 ...

A Tucker hexagon is a hexagon inscribed in a reference triangle that has sides which are alternately parallel and antiparallel to the corresponding sides of the triangle. ...