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The points of tangency of the Lucas inner circle with the Lucas circles are the inverses of the vertices A, B, and C in the Lucas circles radical circle. These form the Lucas ...

The Lucas tangents triangle (a term coined here for the first time) is the triangle DeltaT_AT_BT_C formed by the pairwise tangents of the Lucas circles of a given reference ...

The outer Napoleon circle, a term coined here for the first time, is the circumcircle of the outer Napoleon triangle. It has center at the triangle centroid G (and is thus ...

Triangle picking is the process of picking triangles at random within a given shape in the plane, in space, or in a higher dimension. The most natural definition of a random ...

Let three isoscelizers I_(AC)I_(AB), I_(BA)I_(BC), and I_(CA)I_(CB) be constructed on a triangle DeltaABC, one for each side. This makes all of the inner triangles similar to ...

Brocard's problem asks to find the values of n for which n!+1 is a square number m^2, where n! is the factorial (Brocard 1876, 1885). The only known solutions are n=4, 5, and ...

Let the divisor function d(n) be the number of divisors of n (including n itself). For a prime p, d(p)=2. In general, sum_(k=1)^nd(k)=nlnn+(2gamma-1)n+O(n^theta), where gamma ...

The nth root of the denominator B_n of the nth convergent A_n/B_n of a number x tends to a constant lim_(n->infty)B_n^(1/n) = e^beta (1) = e^(pi^2/(12ln2)) (2) = 3.275823... ...

By way of analogy with the prime counting function pi(x), the notation pi_(a,b)(x) denotes the number of primes of the form ak+b less than or equal to x (Shanks 1993, pp. ...

An elliptic curve of the form y^2=x^3+n for n an integer. This equation has a finite number of solutions in integers for all nonzero n. If (x,y) is a solution, it therefore ...