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The line bisecting a given line segment P_1P_2 can be constructed geometrically, as illustrated above.

The inverse curve of the lituus is an Archimedean spiral with m=2, which is Fermat's spiral.

The catacaustic of a logarithmic spiral, where the origin is taken as the radiant point, is another logarithmic spiral. For an original spiral with parametric equations x = ...

The inverse curve of the logarithmic spiral r=e^(atheta) with inversion center at the origin and inversion radius k is the logarithmic spiral r=ke^(-atheta).

The pedal curve of a logarithmic spiral with parametric equation f = e^(at)cost (1) g = e^(at)sint (2) for a pedal point at the pole is an identical logarithmic spiral x = ...

The radial curve of the logarithmic spiral is another logarithmic spiral.

An equilateral parallelogram whose acute angles are 45 degrees. Sometimes, the restriction to 45 degrees is dropped, and it is required only that two opposite angles are ...

The Lucas central circle is the circumcircle of the Lucas central triangle. The center, radius, and circle function appear to be complicated, and neither the center nor the ...

The MacBeath circle, a term coined here for the first time, is the circumcircle of the MacBeath triangle. It has a fairly complicated radius, center function, and circle ...

The MacBeath triangle DeltaX_AX_BX_C (a term coined here for the first time), is the triangle whose vertices are the contact points of the MacBeath inconic with the reference ...