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Let five circles with concyclic centers be drawn such that each intersects its neighbors in two points, with one of these intersections lying itself on the circle of centers. ...

Given a parabola with parametric equations x = at^2 (1) y = 2at, (2) the negative pedal curve for a pedal point (x_0,0) has equation x_n = (at^2[a(3t^2+4)-x_0])/(at^2+x_0) ...

A hypersphere S^n is parallelizable if there are n vector fields that are linearly independent at each point. There exist only three parallelizable spheres: S^1, S^3, and S^7 ...

The perpendicular bisector of a line segment is the locus of all points that are equidistant from its endpoints. This theorem can be applied to determine the center of a ...

The principal theorem of axonometry, first published without proof by Pohlke in 1860. It states that three segments of arbitrary length a^'x^', a^'y^', and a^'z^' which are ...

Prince Rupert's cube is the largest cube that can be made to pass through a given cube. In other words, the cube having a side length equal to the side length of the largest ...

The maximum and minimum of the normal curvature kappa_1 and kappa_2 at a given point on a surface are called the principal curvatures. The principal curvatures measure the ...

A simple closed curve on a sphere that is not necessarily a great circle but merely intersects as a great circle would (Billera et al. 1999).

The quadrifolium is the 4-petalled rose curve having n=2. It has polar equation r=asin(2theta) (1) and Cartesian equation (x^2+y^2)^3=4a^2x^2y^2. (2) The area of the ...

The radius of curvature is given by R=1/(|kappa|), (1) where kappa is the curvature. At a given point on a curve, R is the radius of the osculating circle. The symbol rho is ...