**ERIC Number:**EJ862209

**Record Type:**Journal

**Publication Date:**2009-Jan

**Pages:**5

**Abstractor:**As Provided

**Reference Count:**0

**ISBN:**N/A

**ISSN:**ISSN-0746-8342

Diametric Quadrilaterals with Two Equal Sides

Beauregard, Raymond A.

College Mathematics Journal, v40 n1 p17-21 Jan 2009

If you take a circle with a horizontal diameter and mark off any two points on the circumference above the diameter, then these two points together with the end points of the diameter form the vertices of a cyclic quadrilateral with the diameter as one of the sides. We refer to the quadrilaterals in question as diametric. In this note we consider diametric quadrilaterals having two equal sides. They can be reduced to the two forms (a, b, a, d) (an isosceles trapezoid) and (b, a, a, d) (a skewed kite). It is shown that these quadrilaterals are diametric if and only if (r, a, d) is a right triangle satisfying d less than [square root]2a where r = [square root](d[superscript 2] - a[superscript 2]), in which case the area of either quadrilateral is ((b+d)/4) [square root](d[superscript 2] - b[superscript 2]), where b = d - 2a[superscript 2]/d. The integer case (Brahmaguta quadrilaterals) is then considered. It is shown that each primitive Pythagorean triple (t, u, v) with v greater than [square root]2u determines a Brahmagupta diametric quadrilateral with two equal sides uv, two other sides v[superscript 2] - 2u[superscript 2], v[superscript 2], and area ut[superscript 3]. Moreover every primitive diametric quadrilateral with integer sides, two of which are equal, and with integer diagonals, arises in this way.

Descriptors: Geometric Concepts, College Mathematics, Mathematics Instruction, Mathematical Concepts, Geometry, Mathematical Logic, Validity, Equations (Mathematics), Algebra

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**Publication Type:**Journal Articles; Reports - Descriptive

**Education Level:**Higher Education

**Audience:**N/A

**Language:**English

**Sponsor:**N/A

**Authoring Institution:**N/A