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A tree is planted at each lattice point in a circular orchard which has center at the origin and radius r. If the radius of trees exceeds 1/r units, one is unable to see out ...

The centroid of the four points constituting an orthocentric system is the center of the common nine-point circle (Johnson 1929, p. 249). This fact automatically guarantees ...

Let pi be a unitary representation of a group G on a separable Hilbert space, and let R(pi) be the smallest weakly closed algebra of bounded linear operators containing all ...

A subset X of R^n is star convex if there exists an x_0 in X such that the line segment from x_0 to any point in X is contained in X. A star-shaped figure is star convex but ...

The figure formed when the midpoints of adjacent sides of a quadrilateral are joined. Varignon's theorem demonstrated that this figure is a parallelogram. The center of the ...

If each of two curves meets the line at infinity in distinct, nonsingular points, and if all their intersections are finite, then if to each common point there is attached a ...

The Yiu circle is the circumcircle of the Yiu triangle DeltaO_AO_BO_C. It has center function given by alpha=af(a,b,c), where f(a,b,c) is a 14th-order polynomial. Its radius ...

Through a point K in the plane of a triangle DeltaABC, draw parallelians through a point as illustrated above. Then there exist four points K for which ...

To generate random points over the unit disk, it is incorrect to use two uniformly distributed variables r in [0,1] and theta in [0,2pi) and then take x = rcostheta (1) y = ...

Two metrics g_1 and g_2 defined on a space X are called equivalent if they induce the same metric topology on X. This is the case iff, for every point x_0 of X, every ball ...