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An (infinite) line determined by two points (x_1,y_1) and (x_2,y_2) may intersect a circle of radius r and center (0, 0) in two imaginary points (left figure), a degenerate ...

The involute of the circle was first studied by Huygens when he was considering clocks without pendula for use on ships at sea. He used the circle involute in his first ...

The diagonal triangle of a complete quadrangle is the triangle formed by its three diagonal points. If the quadrangle is a cyclic quadrilateral, then the circle is the polar ...

Let O and I be the circumcenter and incenter of a triangle with circumradius R and inradius r. Let d be the distance between O and I. Then d^2=R(R-2r) (Mackay 1886-1887; ...

One of the beautiful arrangements of circles found at the Temple of Osiris at Abydos, Egypt (Rawles 1997). The pattern also appears in Phoenician art from the 9th century BC ...

A golden isozonohedron is a zonohedron all of whose faces are golden rhombi. There exist exactly five golden isozonohedra, as summarized in the following table. face count ...

A Greek cross, also called a square cross, is a cross in the shape of a plus sign. It is a non-regular dodecagon. A square cross appears on the flag of Switzerland, and also ...

Given collinear points W, X, Y, and Z, Y and Z are harmonic conjugates with respect to W and X if (|WY|)/(|YX|)=(|WZ|)/(|XZ|). (1) W and X are also harmonic conjugates with ...

Any one of the eight Apollonius circles of three given circles is tangent to a circle H known as a Hart circle, as are the other three Apollonius circles having (1) like ...

Two figures are homothetic if they are related by an expansion or geometric contraction. This means that they lie in the same plane and corresponding sides are parallel; such ...