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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.