Download Triangles with Collinear Circumcenters

Project AMP
Dr. Antonio R. Quesada – Director, Project AMP
Triangles with Collinear Circumcenters
Lesson Summary:
This lesson has students examine a sequence of triangles with collinear
circumcenters and similar to a given triangle. The circumcenter of a triangle is the
center of the circumscribed circle and is found by finding the intersection of the
perpendicular bisectors of the sides of a triangle. This would be a great extension to a
lesson on circumcenter.
Key Words:
Circumcenter, perpendicular bisectors
Background knowledge:
Students need to be familiar with Cabri Geometry. They also need to understand
similar triangles and the meaning of circumcenter.
Learning Objectives:
Students will identify the circumcenter of a triangle and explore the relationship
between triangles will collinear circumcenters.
Materials:
Cabri Geometry
Procedures:
Pass out worksheets to students. Organize them into groups of two or three. Have
them work on the activities using geometry software.
Project AMP
Dr. Antonio R. Quesada – Director, Project AMP
Triangles with Collinear Circumcenters
Team members’ names: _________________________________________________
File names: ____________________________________________________________
Goal: Construct a sequence of triangles with collinear circumcenters and similar to a
given triangle.
1) Draw a triangle ABC
2) Find the midpoint of the sides AB, AC, and BC.Label them as E, F, and G.
3) Draw perpendicular bisectors to the sides AB, BC, and AC and observe that they
intersect each other at a single point. (Circumcenter)
4) Make this intersection in blue color and label it C1
5) Construct the triangle EGF from the midpoints E, F, and G.
6) Make a chart showing the ratio AB/FG, AC/EG, and BC/EF.
7) What do you observe from this chart?
8) Considering step 7, guess the ratio of the areas of the triangles ABC and EGF, and
verify your guess by measuring the areas.
9) Measure the angles ∠ A & ∠ F, also ∠ B &∠ G, and ∠ C & ∠ E.
What is the relation between each pair of angles?
Project AMP
Dr. Antonio R. Quesada – Director, Project AMP
10) Are the triangles ABC & EGF similar?
11) Repeat steps 2,3, and 4 for the triangle EGF to construct a new triangle HIJ and its
cicumcenter C2 that is colored in blue. (Make a macro with initial object segments AB,
AC, and BC and final object segments EG, EF, FG, and the circumcenter)
12) Repeat steps 11 for the triangle HIJ to construct a new triangle and its circumcenter
C3 that is colored in blue.
13) What do you observe regarding the blue points?
14) Move any of vertices A, B, or C. Does the colinearity remain unchanged?
15) What is the result of repeating this process to new triangles infinitely many times?
16) Are these points approaching a known center of the triangle ABC?
Extension Does this lab provide the same result using angle bisectors(incenter),
altitudes(orthocenter), or medians(centroid)?
Project AMP
Dr. Antonio R. Quesada – Director, Project AMP
Project AMP
Dr. Antonio R. Quesada – Director, Project AMP
Project AMP
0
Dr. Antonio R. Quesada – Director, Project AMP
Project AMP
Dr. Antonio R. Quesada – Director, Project AMP