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Four circles c_1, c_2, c_3, and c_4 are tangent to a fifth circle or a straight line iff T_(12)T_(34)+/-T_(13)T_(42)+/-T_(14)T_(23)=0. (1) where T_(ij) is the length of a ...

Let X_1,X_2,...,X_N be a set of N independent random variates and each X_i have an arbitrary probability distribution P(x_1,...,x_N) with mean mu_i and a finite variance ...

An equation is said to be a closed-form solution if it solves a given problem in terms of functions and mathematical operations from a given generally-accepted set. For ...

If P is a pedal point inside a triangle DeltaABC, and P_A, P_B, and P_C are the feet of the perpendiculars from P upon the respective sides BC, CA, and AB, then ...

Draw lines P_AQ_A, P_BQ_B, and P_CQ_C through the symmedian point K and parallel to the sides of the triangle DeltaABC. The points where the parallel lines intersect the ...

Gaussian elimination is a method for solving matrix equations of the form Ax=b. (1) To perform Gaussian elimination starting with the system of equations [a_(11) a_(12) ... ...

A geometric construction done with a movable compass alone. All constructions possible with a compass and straightedge are possible with a movable compass alone, as was ...

The lines containing the three points of the intersection of the three pairs of opposite sides of a (not necessarily regular) hexagon. There are 6! (i.e., 6 factorial) ...

A rational number is a number that can be expressed as a fraction p/q where p and q are integers and q!=0. A rational number p/q is said to have numerator p and denominator ...

A special function is a function (usually named after an early investigator of its properties) having a particular use in mathematical physics or some other branch of ...