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All the propositions in projective geometry occur in dual pairs which have the property that, starting from either proposition of a pair, the other can be immediately ...

Integrals over the unit square arising in geometric probability are int_0^1int_0^1sqrt(x^2+y^2)dxdy=1/3[sqrt(2)+sinh^(-1)(1)] int_0^1int_0^1sqrt((x-1/2)^2+(y-1/2)^2)dxdy ...

Let f:R×R->R be a one-parameter family of C^3 maps satisfying f(0,0) = 0 (1) [(partialf)/(partialx)]_(mu=0,x=0) = -1 (2) [(partial^2f)/(partialx^2)]_(mu=0,x=0) < 0 (3) ...

The subset B of the Euclidean plane formed by the union of the interval [0,1] of the x-axis and all line segments of unit length passing through the origin which form an ...

Given a unit circle, pick two points at random on its circumference, forming a chord. Without loss of generality, the first point can be taken as (1,0), and the second by ...

O_h is the point group of symmetries of the octahedron having order 48 that includes inversion. It is also the symmetry group of the cube, cuboctahedron, and truncated ...

Given a circle expressed in trilinear coordinates by a central circle is a circle such that l:m:n is a triangle center and k is a homogeneous function that is symmetric in ...

Consider the distribution of distances l between a point picked at random in the interior of a unit cube and on a face of the cube. The probability function, illustrated ...

The evolute of an ellipse specified parametrically by x = acost (1) y = bsint (2) is given by the parametric equations x_e = (a^2-b^2)/acos^3t (3) y_e = (b^2-a^2)/bsin^3t. ...

For a nonzero real number r and a triangle DeltaABC, swing line segment BC about the vertex B towards vertex A through an angle rB. Call the line along the rotated segment L. ...