### Geometry Forum Summer Institute - July 9-15, 1995

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### Daily Summary Monday, July 10, 1995

Monday morning was fun for those of us here at Swarthmore College. After a more interactive Connections, the participants once again hit the computer lab. Some time was spent checking e-mail, surfing the Web, and polishing home pages.

The primary activity was to explore the new features of The Geometer's Sketchpad, version 3.0. Before starting, Steve reminded us of the true value of Sketchpad. Some teachers have asked why, for example, it doesn't provide the user with a tool to draw rectangles or other polygons. Well, that's exactly the point. It's not a drawing program, it's a geometry tool. If (and only if) you know the geometry, you can draw virtually anything.

Annie gave a quick lesson on how to use the new additions available in 3.0. We drew an equilateral triangle, saved it as a script, and then stored that script in a folder. After doing this, you can access a new tool (at the bottom of the tool menu) that plays back the scripts you've saved. So, in the case of the equilateral triangle, you now have a tool that draws a new triangle anywhere on your sketch.

The script tool also comes equipped with a wealth of constructions that are built into Sketchpad. Another new item that is the dialogue box at the bottom of the screen which tells you exactly what you're doing as you manipulate your sketch.

Our last activity was to use Sketchpad to try to solve the following problem, as submitted by Annie:

From: annie@mathforum.org (Annie Fetter)
Subject: Napoleon's Theorem

1. Construct a triangle, any triangle. Construct equilateral triangles on the sides of the triangles. Connect the centers (centroids) of these triangles. What is true?

2. Connect the outside vertices of the equilateral triangles with the opposite vertex of the original triangle. What is true of the line segments?

(Paul and Steve deserve particular praise for finding all four answers to this question.)

3. What happens when you reflect each centroid over the closest edge of your original triangle ABC? What is the difference in areas between the outer and inner Napoleanic triangles?

4. How are these line segments (or how is one cool thing about them) related to the original triangle?

Have fun.

Eric Sasson
Geometry Forum