Each ellipse and hyperbola has a circle associated with it called the director circle. In this article, the author derives the equations of the circle for the ellipse and hyperbola through a different approach. Then the author concentrates on the director circle of the central conic given by the general quadratic equation. The content of this…

The Greek astronomer Ptolemy of Alexandria (second century) and the Indian mathematician Brahmagupta (sixth century) each have a significant theorem named after them. Both theorems have to do with cyclic quadrilaterals. Ptolemy's theorem states that: In a cyclic quadrilateral, the product of the diagonals is equal to the sum of the products of two…

The topic of orthogonal trajectories is taught as a geometric application of first order differential equations. Instructors usually elaborate on the concept of a family of curves to emphasize that they are different even if their members are of the same type. In this article the author considers five families of ellipses, discusses their…

The sequence 1, 1, 2, 3, 5, 8, 13, 21, ..., known as Fibonacci sequence, has a long history and special importance in mathematics. This sequence came about as a solution to the famous rabbits' problem posed by Fibonacci in his landmark book, "Liber abaci" (1202). If the "n"th term of Fibonacci sequence is denoted by [f][subscript n], then it may…

In the seventh century, around 650 A.D., the Indian mathematician Brahmagupta came up with a remarkable formula expressing the area E of a cyclic quadrilateral in terms of the lengths a, b, c, d of its sides. In his formula E = [square root](s-a)(s-b)(s-c)(s-d), s stands for the semiperimeter 1/2(a+b+c+d). The fact that Brahmagupta's formula is…

In this article, the author takes up the special trinomial (1 + x + x[squared])[superscript n] and shows that the coefficients of its expansion are entries of a Pascal-like triangle. He also shows how to calculate these entries recursively and explicitly. This article could be used in the classroom for enrichment. (Contains 1 table.)