Nathanael Honkomp Lesson title: The Nine-Point Circle Lesson ...

Nathanael HonkompLesson title: The Nine-Point CircleLesson Objectives:1. For the Students to construct the nine-point circle, and review the ideas ofcircumcircle, orthocenter, midpoints, altitudes, and the perpendicular bisectortheorem.2. Introduce the Euler and Simson Lines through construction activities.3. Show the students there is more than one way to solve a problem.4. display the interrelatedness of various geometrical figures and entities.Materials: A computer lab or a set of calculators equipped with Cabri Geometry II. Labmanuals for the students to follow and answer lab questions on.Procedure:1. Group students into lab teams by method of choice.2. Hand out lab manuals and let the students work through them, and assist themwith questions when they get stuck.Assessment: View students completed lab drawings, and collect lab manuals to see thatquestions were properly graded.Disclaimer: If the students are unfamiliar with Cabri Geometry, they can use the helpmenu for an explanation of what each tool does.

Lab 1a: Constructing a circumcircle1. Draw a triangle ? ABC. (Triangle Tool)2. Place midpoints on each side of the triangle. (Midpoint Tool)3. Draw a lines perpendicular to each side through these points. What can you sayabout these lines in regard to the sides of the triangles? (Perpendicular Tool)_________________________________________________________________4. Label the point where all three lines intersect point CC. (Label Tool)5. Since point CC is concurrent to all three perpendicular bisectors, whatconclusions can you draw about its distance from each of the three vertices, andwhy?____________________________________________________________________________________________________________________________________6. Draw a circle with center point CC, and radius point A, B, or C. Drag each of thetriangle vertices around the page to show that the circle does indeed intersect allthree points. This is called a circumcircle of a triangle. Why do you suppose it iscalled a circumcircle?__________________________________________________________________

Lab 1b: Constructing the Nine-Point Circle1. Construct a triangle ? ABC (Triangle Tool)2. Create midpoints L, M, and N on segments AB , BC , and CD respectively.(Midpoint Tool)3. Draw lines through vertices A, B, and C perpendicular to the opposite sides.(Perpendicular Tool)4. Create points D, E, and F at the intersections of the perpendicular lines andtriangle sides AB, BC, and CA respectively. Create a point H where the threelines intersect, this point is called the orthocenter of the triangle. (Point Tool)suur suur suur5. Hide lines AE , CD , and BF . (Hide/Show Tool)6. Draw segments AH , BH , and CH . (Segment Tool)7. Create and label the midpoints on each of the new segments X, Y, and Zrespectively. (Midpoint Tool)

8. Using the information presented in the circumcircle lab, create a circumcircleabout triangle ? ABC.9. Create a segment HCC . (Segment Tool) (Note: this segment is part of what iscalled the Euler Line, which is a line established by the centroid of a triangle, theorthocenter, the circumcenter, and the nine-point center.)10. Draw and label the midpoint of this new segment point U. (Midpoint Tool)11. Draw a circle with center point U and radius point L. (Circle Tool)12. How can you check to make sure that all nine points lie on the circle?_______________________________________________________________________________________________________________________________________________________________________________________________________________

Lab 2: Constructing the nine-point circle as a locus1. Construct a triangle ? ABC. (Triangle Tool)2. Draw the three altitudes of the triangle. (Perpendicular Tool)3. Label the feet of the altitudes points D, E, and F, and label the orthocenter ofthe triangle point H. (Label Tool)4. Create and label the mid points of the segments between the vertices of thetriangle and the orthocenter as points X, Y, and Z. (Midpoint Tool)5. Create midpoints L, M, and N for each side of the triangle respectively.(Midpoint Tool)6. These points are the nine points of the nine-point circle.7. Draw the perpendicular bisector to each side of the triangle. (PerpendicularTool)

8. Label the point where they intersect CC, and create a circle using CC as thecenter and any of the triangles vertices as the radius point. (Circle Tool)9. For ease of viewing, hide point CC, and the perpendicular bisector lines.(Hide/Show Tool)10. Place a point P anywhere on circle CC. (Point Tool)11. Create lines that are collinear to each of the three sides of the triangle. (LineTool)12. Draw Perpendiculars to these lines through the point P, and place points ateach of the newly created intersections. (Perpendicular & Point Tools)13. Draw a line segment between the two perpendicularity points the farthest fromeach other. What do you notice about the third point? Check to see if yourobservation is true, is it? (Segment Tool, Collinear Tool)_______________________________________________________________

14. This line is called the Simson Line. Now hide the lines created in steps 11and 12, and hide point CC. (Hide/Show Tool)15. Create a segment PH , and its midpoint. What does the Simson Line do tothis segment? (Segment & Midpoint Tools)______________________________________________________________________________________________________________________________16. Create a locus between the midpoint of segment PH and point P. What doyou notice about the locus that is created in relation to the nine points created insteps 1-5? Drag point P around the circle to check your observation, does yourobservation hold true?______________________________________________________________________________________________________________________________________________________________________________________________________17. Describe what the circle created in this drawing is a locus of._____________________________________________________________________________________________________________________________________________________________________________________________