Area, Diagonals, and Circumcircle of a Cyclic Quadrilateral

Autor/in	Ayoub, Ayoub B.
Titel	Area, Diagonals, and Circumcircle of a Cyclic Quadrilateral
Quelle	In: Mathematics and Computer Education, 40 (2006) 1, S.58-63 (6 Seiten) PDF als Volltext Verfügbarkeit
Sprache	englisch
Dokumenttyp	gedruckt; online; Zeitschriftenaufsatz
ISSN	0730-8639
Schlagwörter	Geometric Concepts; Mathematical Formulas; Mathematics Education; Mathematics Instruction; College Mathematics; Problem Solving; Computation; History + Suchen Sie Ihr Suchwort? Elementare Geometrie; Mathematische Formel; Mathematische Bildung; Mathematics lessons; Mathematikunterricht; Problemlösen; Geschichte; Geschichtsdarstellung
Abstract	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 symmetric in a, b, c, d suggests that there are different cyclic quadrilaterals having the same area and the same side lengths, though the sides may be in different orders. Since the vertices of a cyclic quadrilateral lie on a circle, then actually there are only three orders for the sides, namely (a, b, c, d), (a, b, d, c), and (a, c, b, d), where a is opposite to c, d and b respectively. The author will show that the three quadrilaterals, corresponding to these orders, can be inscribed in the same circle. He will calculate the lengths of their diagonals and the radius of their circumcircle. Before he does that, he will furnish a proof for Brahmagupta's formula. He applies basic geometric and trigonometric results which would make the content of this article usable in the classroom. (Contains 3 figures.) (ERIC).
Anmerkungen	MATYC Journal Inc. Mathematics and Computer Education, P.O. Box 158, Old Bethpage, NY 11804. Tel: 516-822-5475; Web site: http://www.macejournal.org
Erfasst von	ERIC (Education Resources Information Center), Washington, DC
Update	2017/4/10