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The angle bisector of an angle in a triangle divides the opposite side in the same ratio as the sides adjacent to the angle.

Find necessary and sufficient conditions that determine when the integral curve of two periodic functions kappa(s) and tau(s) with the same period L is a closed curve.

The map x^' = x+1 (1) y^' = 2x+y+1, (2) which leaves the parabola x^('2)-y^'=(x+1)^2-(2x+y+1)=x^2-y (3) invariant.

A Mycielski graph M_k of order k is a triangle-free graph with chromatic number k having the smallest possible number of vertices. For example, triangle-free graphs with ...

Points, also called polar reciprocals, which are transformed into each other through inversion about a given inversion circle C (or inversion sphere). The points P and P^' ...

Let a convex polygon be inscribed in a circle and divided into triangles from diagonals from one polygon vertex. The sum of the radii of the circles inscribed in these ...

Line segment picking is the process of picking line segments at random within a given shape in the plane, in space, or in a higher dimension. The most natural definition of a ...

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 ...

For a cyclic quadrilateral, the sum of the products of the two pairs of opposite sides equals the product of the diagonals AB×CD+BC×DA=AC×BD (1) (Kimberling 1998, p. 223). ...

The Gauss-Bonnet formula has several formulations. The simplest one expresses the total Gaussian curvature of an embedded triangle in terms of the total geodesic curvature of ...