Lesson 3: Parallelogram, Circle, Vertices




Student Audience and Timeline:

            This lesson is designed for implementation in a high school Euclidean Geometry course and is designed to be completed in one 55 minute class period. 



á      Students will construct a dynamic sketch of a parallelogram

á      Students will construct circles through each of 4 pairs of adjacent vertices

á      Students will investigate the results of connecting the centers of these four circles, forming conjectures about resulting figures and noting special cases

á      Students will articulate their conjecture and provide a supporting proof, either formal or informal

á      Students will extend this outlined investigation to other polygons of their choice, creating an associated sketch and forming conjectures about their investigation


Mathematical Concepts:

á      Properties of a parallelogram

á      Construction of a circle through three points

á      Parallel and perpendicular lines

á      Perpendicular bisector of a segment

á      Midsegment


Lesson Synopsis:

            The lesson will begin with an instructor-initiated student investigation of the results of connecting the centers of the circles passing through each set of three adjacent vertices of a parallelogram.  During this investigation, students will construct a dynamic sketch then form and support conjectures.  The second component of this lesson is a student-selected extension of the initial investigation.  Students, either working individually or in pairs should fully investigate a related but different mathematical situation, creating a dynamic sketch and supporting their findings.  The results of these extensions will be compiled as a Powerpoint slideshow, with each student (or student pair) contributing one or two slides of their extension exploration. 





á      GeometerŐs Sketchpad software

á      Microsoft Powerpoint software, or similar



            The instructor will begin the lesson by leading a brief discussion about constructing a circle through three given points.  The instructor may highlight the elements of this construction, connecting this to constructing the circumcircle of a triangle.  In particular, the instructor should question the students about the possibility of creating a circle that passes through any three points.  The instructor will then shift the lesson focus toward constructing circle through sets of three special points, namely the four sets of three adjacent vertices of a parallelogram. 

            After the instructor poses the question of what these circle may look like and what properties their centers may possess, students will work either individually or in small groups to construct a dynamic GeometerŐs Sketchpad sketch.  A sample sketch can be found here.  Students should investigate the properties of resulting circle as related to parallelogram type (e.g. square, rectangle, rhombus, general parallelogram).  Additionally, students will investigate the results of connecting the centers of the four circles in their sketch.  After making conjectures about the resulting figure and noting special degenerate cases, students will develop supported reasoning, either as a formal or informal proof, explaining their conjectures. 

            Next, students will design an extension of this investigation.  Some may choose to look at a broader range of original quadrilaterals (including kites, trapezoids, and irregular quadrilaterals).  Others may want to investigate connecting the centers of circle created from polygons with different numbers of sides.  As students investigate, the instructor will encourage them to select one piece of their work that they find particularly interesting and develop a single (or set of two) Powerpoint slides that captures their investigation and findings.  Students should strive to incorporate images, dynamic sketches and a clearly articulated summary of this component of their investigation.  The studentsŐ slides will then be compiled in a logical order for presentation.


Facilitator Questions:

á      In order to connect the centers of the circles, is it necessary to construct each circle?

á      Can you find a shortcut construction that yields the same final figure (from connecting the circle centers)?

á      What properties does the resulting figure appear to have?

á      How can you use GeometerŐs Sketchpad to support your conjectures about the quadrilateralŐs properties?

á      How does the resulting quadrilateral relate to the original parallelogram?

á      Are there any situations in which the resulting quadrilateral is very different or very unique? When does this occur?

á      How does the area of the parallelogram and the resulting quadrilateral relate?

á      How does the orientation of the original parallelogram and resulting quadrilateral relate?

á      Is the resulting quadrilateral every wholly contained within the original parallelogram?  When does this occur?