Analytic Geometry Part 1 Unit 2 Project - Napoleon's Theorem Overall Expectations: Model and solve problems involving intersection of two straight lines Solve problems using analytic geometry involving properties of lines and line segments Verify geometric properties of triangles and using analytic geometry Specific Expectations: . Determine thorough investigation, some characteristics and properties of geometric figures (Medians of a triangle, similar figures constructed on the sides of a triangle) Napoleon's Theorem: The centroid of a triangle is also the triangle's centre of Mass or "balance point". This fact can be used to design a mobile or hanging sculpture. If a string is attached at the Centroids, the triangle will hang horizontally. When a complex shape is used, in balance point can be found by first locating the cencroids of all the triangle that can make up the shape. Supposed that you are using a piece of graph paper to make a star-shaped mobile by constructing equilateral triangle on the exterior sides of an existing triangle ABC. a. The vertices of a triangle ABC are A(3, 1), B(-4, -3), and C(6, -1). Construct an equilateral triangle on each exterior side of triangle ABC. b. Find the coordinates of the centroid of each equilateral triangle. c. Show that the centroids form the vertices of a new equilateral triangle. d. Find the coordinates of the centroid of this triangle. e. Construct the start using rigid construction paper or cardboard. Use the shape to verify whether or not the centroid of the new triangle is the balance point for the start. f. Write the summary of YOUR findings. g. Write a small report about Napoleon a Mathematician that include Napoleon theorem as well