1. Click the Segment between two points tool, and click two distinct places on the drawing pad to construct segment AB. 2. To construct a circle with center A passing through B, click the Circle with Center through Point tool, click point A, then click point B. 3. To construct another circle with center B passing through A, with the Circle with Center though Point still active, click point B and then click point A. 4. Next, we have to intersect the circles. To intersect the two circles, click the inverted triangle on the New Point tool, select Intersect Two Objects, then click the circumference of both circles. Notice that two points will appear in their intersections. 5. We only need three points, points A, B and C, to form an equilateral triangle, so we will hide the two circles, segment AB and point D. To do this, right click each object and click the Show Object option to uncheck it. In hiding segment AB, be sure that you do not click points A or B 6. With only three points remaining on the drawing pad, click the Polygon tool and click the points in the following order: point A, point B, point C and then point A to close the polygon. 7. Using the Move button, move the vertices of the triangle. What do you observe? 8. You have probably observed that it seems that ABC is an equilateral triangle. In fact, it is. To verify, we can display the interior angles of the triangle. To do this, click the Angle tool, then click the interior of the triangle. 9. What do you observe? Move the vertices of the triangle. Is your observation still the same? 10. Now construct midpoints of each line segment by clicking on the Midpoint or Center tool and clicking on point A and then point B, point B and then point C, point C and then point A. 11. Connect the midpoints with the segment tool. How many triangles are inside the original triangle now? 12. Are all of these triangles equilateral as well? Check the angle sizes on all of these triangles by following step 8. 13. Test the theorem: “A triangle is equilateral if and only if any three of the smaller triangles have either the same perimeter.” 14. To verify the length of the sides use the Properties window. To do this, right click one of the sides of the triangle, click Object Properties from the context menu. 15. In the Object Properties window, select the Basic tab. Be sure that the Show label check box is checked and choose Value from the Show label drop down list box. 16. Complete with all other sides of the triangle. 17. Are the side lengths the same? Does this prove the theorem?