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Zachary Tocchi
Dr. Antonio Quesada
Concepts in Geometry
Activity Log
Intro to Cabri
Lab 1
This lab presents a classic “handshakes” problem and solves it using the Cabri software and
geometry. Students are to create a circle with points on the circle which represent people. Then the
students must draw lines from each of the points to each of the other points until there are no more
points that are not connected with lines.
Lab 2
This lab was all about performing transformations on shapes using the Cabri software. Students
will also use the Cabri software to calculate area of shapes and see how the area changes when the
shape changes or when the shape moves around on the coordinate plane.
Lab 3
This lab uses a famous mathematicians’ theorem and proves it by picture. Students will create a
circle and place a point in it, then draw two segments through the point touching each end of the circle.
Using the Cabri software, students will discover something very interesting about the two lines.
Lab 4
This lab teaches students how to create a macro in Cabri. One point that is important to note is
that all macros created should have a “F1” description to them; meaning that, when I press F1, a
description of the macro pops up.
Euclidean Geometry
Circumcenter
This activity teaches students to find the circumcenter of a triangle. This can be done by using
perpendicular bisectors of two sides of a triangle. Where the lines intersect is the circumcenter of the
triangle. Next, students create a circumcircle using the circumcenter and the three vertices of the
triangle.
Angle Bisector
This activity has two parts. The first has students create an angle bisector using a piece of paper
and a protractor. Students can see how an angle bisector is formed and then can be shown that it also
applies to other angles besides right angles. This leads to the second part of the activity where students
create a random size angle and create an angle bisector of it. I personally would like to be taught how to
create an angle bisector by hand.
Incenter
This activity walks students through how to find the incenter and incircle of a triangle using the
Cabri Software. I would be interested to see a prequel to this lab where students try to find these
properties of a triangle with pen and paper.
Centroid
The centroid activity shows students how to find the center of mass by using medians of a
triangle. The first activity has students find the center of mass using pencil and straight edge. The second
activity has them complete the same task using the Cabri software.
Orthocenter
The orthocenter activity has students using the Cabri software to explore the properties of the
altitudes of a triangle, and the point where the altitudes meet in the triangle. Students will explore
obtuse, right, and acute triangles.
Modern Euclidean Geometry
Ceva’s Theorem
This activity has students create a triangle and demonstrates what exactly Ceva’s theorem is. At
the end of the activity, it has the student define Ceva’s theorem in their own words.
Menelaus’ Theorem
This activity has students discover Menelaus’ Theorem using the interactive geometry software,
Cabri.
Nine Point Circle
The Nine-Point Circle lab has students construct a nine-point circle in a triangle using the
essential nine points (which they discover in the lab). It also has students compare this circle to the
circumcircle of the triangle and how the radii compare.
Taxicab Geometry
Taxicab Geometry
This activity introduces students to the concept of the Taxicab geometry then applies the
concepts to shapes which include triangles, similar triangles, squares, circles, ellipses, hexagon and other
various polygons. Students are supposed to generalize the findings of each shape and see what shape
they act like (eg. A square is the Taxicab circle)
Tessellations
Tessellations
This lab has students discover which regular shapes are tessellatable… if that is even a word?
Combinatorial Theorems in Geometry
Euler’s Theorem
By creating polyhedra, students discover how the number of vertices, edges, and faces relate.
The number 2 comes up a lot!!
Pick’s Theorem
By creating various polygons, students discover the relationship between vertices, edges, and
the shape’s area.
Class Presentations
The Napoleon Points
In this activity, students are guided on the construction of the two Napoleon points and are then
asked several questions about the relationship between the points. These are different centers of a
triangle that were not discussed earlier in the class.
The Schiffler Point
This center of a triangle uses the angle bisectors of the three angles of a triangle, and calculates
the Euler lines of each of the triangles. We discover in this activity, that the four Euler lines meet at one
point.
Hyperbolic Geometry
In this activity, students use Cabri with a hyperbolic add-on to construct various figures in
hyperbolic geometry. Such activities include circumcenter of d-triangles, properties of a circle in
hyperbolic space, parallel lines, perpendicular lines, and isosceles triangle theorem, and a few other
properties.
Inversion
In this activity, students find the inverse of a point in a circle, properties of inversion for a line
and a line through the center of the circle, the inverse of a circle in numerous positions, properties of
orthogonal circles, and the preservation of angles.
Platonic Solids
In this activity, students use Cabri 3D to construct a regular tetrahedron, finding an octahedron,
discover duals, and a few more questions about platonic solids. This lab is NOT complete as we do not
have access to Cabri 3D.