# A Summary of Circumscribable Quadrilaterals

Topics: Triangle, Quadrilateral, Circle Pages: 20 (5315 words) Published: February 23, 2012
CHAPTER 1
INTRODUCTION
1.1 Introduction
Geometry is one of the most interesting fields of mathematics. From the ancient times of the Greeks up to now, it has held captive the imagination of many mathematicians, artists, scientists, engineers and architects. Its application to modernization and technological advancement cannot be denied. Thus, it must be given emphasis in educational institutions particularly in secondary schools. The low achievement test results in mathematics of high school students in the Philippines have always been the subject of study in researches and the center of discussion in conferences. Different strategies and recourses have been drafted to be implemented in schools towards a better performance in the said subject. One such strategy is exposure to varied problems and concepts that cater to higher order thinking skills of students for them to develop critical and analytical skills.

The quadrilateral is a very interesting concept for it is rich in content and applications. Quadrilaterals such as the golden rectangle and cyclic quadrilateral are famous for their intricate properties. In high schools, however, this concept is not given emphasis and teachers are confined within the types of quadrilaterals enumerated in the textbook. In effect, students’ learning is shallow. Their interests are not stimulated and they think quadrilaterals are very simple. But when given problems on quadrilaterals that are of moderate difficulty, they are stumped.

1.2 Statement of the Problem
The aims of this paper are to define circumscribable quadrilaterals and to present the proofs of its properties. It also intends to discuss and prove the conditions for a quadrilateral to be circumscribable.

This paper is inspired by Charles Worrall’s article entitled “ A Journey with Circumscribable Quadrilaterals” published in the Delving Deeper section of the Mathematics Teacher in October 2004. In the paper, the author explained the importance of inquiry in teaching and learning mathematics. He also emphasized creative thinking in order to better understand and appreciate the beauty of mathematics particularly geometry.

1.3 Scope and Limitation of the Study
Due to the limited time devoted in making this paper, it is confined only to the definition of circumscribable quadrilaterals, its properties, and the conditions for its existence. Also, we assume that all quadrilaterals mentioned in this paper are convex quadrilaterals. Furthermore, proofs are done within the confines of Euclidean geometry although projective geometry and trigonometry could also be used. Finally, all figures are constructed using the mathematics software, The Geometer’s Sketchpad.

CHAPTER 2
PRELIMINARIES

This chapter aims to discuss the definitions, postulates, and theorems needed to understand the concepts on circumscribable quadrilaterals. Illustrations and examples are also given to provide a clearer idea of the topic being discussed.

2.1 CIRCLES
Definition 2.1.1.A circle is the set of all points in the plane that are a given distance from a given point. The given point is the center of the circle, and the given distance is the radius. A segment that joins the center to a point on the circle is also called radius. (The plural of radius is radii.) Illustration:

A circle may be named using one capital letter that is its center. For example, a circle whose center is at P will be called circle P.

Definition 2.1.2.A point is inside (in the interior of) a circle if its distance from the center is less than the radius.

Illustration:

Point A is in the interior of circle O.
Definition 2.1.3.A point is outside (in the exterior of) a circle if its distance from the center is greater than the radius. Illustration:

Point C is in the exterior of circle O.
Definition 2.1.4.A tangent line to a circle is a line that intersects the circle at exactly one point. This point is called the point of tangency or point of...