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Eliminate each knot crossing by connecting each of the strands coming into the crossing to the adjacent strand leaving the crossing. The resulting strands no longer cross but ...

The symmedial circle is the circumcircle of the symmedial triangle. It has circle function l=(bc(a^4-a^2b^2-b^4-a^2c^2-b^2c^2-c^4))/(2(a^2+b^2)(a^2+c^2)(b^2+c^2)), (1) which ...

The tangential mid-arc circle is the circumcircle of the tangential mid-arc triangle. Its center and radius appear to be very complicated functions. Its center is not in ...

An inscribed angle in a semicircle is a right angle.

The third Lemoine circle, a term coined here for the first time, is the circumcircle of the Lemoine triangle. It has center function alpha=(f(a,b,c))/a, (1) where f(a,b,c) is ...

The triangle transformation principle gives rules for transforming equations involving an incircle to equations about excircles.

The triquetra is a geometric figure consisting of three mutually intersecting vesica piscis lens shapes, as illustrated above. The central region common to all three lenses ...

A class of area-preserving maps of the form theta_(i+1) = theta_i+2pialpha(r_i) (1) r_(i+1) = r_i, (2) which maps circles into circles but with a twist resulting from the ...

A disk with radius 1. The (open) unit disk can also be considered to be the region in the complex plane defined by {z:|z|<1}, where |z| denotes the complex modulus. (The ...

The integral of 1/r over the unit disk U is given by intint_(U)(dA)/r = intint_(U)(dxdy)/(sqrt(x^2+y^2)) (1) = int_0^(2pi)int_0^1(rdrdtheta)/r (2) = 2piint_0^1dr (3) = 2pi. ...