**Introduction**

Wingeom is a dynamic geometry software created by Philip Exeter University. It is capable of 2-dimensional and 3-dimensional geometric drawing and construction.

This is the first tutorial of the Wingeom Tutorial Series. Most of the construction in this tutorial series will deal with 2-dimensions.

**The Wingeom Environment**

When you open Wingeom, the window shown below will appear. You have to click the** Window** menu, then choose the environment that you want to display. Wingeom can construct figures in Euclidean, hyperoblic and spherical plane.

Figure 1 – The Wingeom window.

It is also capable of constructing Voronoi diagrams and tessellations.

**Using Wingeom in Exploring the Midline Theorem**

In this construction, we will explore the relationship of a triangle and its midline (or midsegment), the segment connecting the midpoints of its two sides as shown in Figure 2.

Figure 2 – Triangle ABC with midline DE.

In the construction below, we will construct 3 points A, B and C and connect them with the *segment *tool. After drawing the triangle, we will get the midpoints of *AB *and *AC* and explore the length and interior angles of the two triangles formed.

To perform the construction, follow the construction steps below and answer the questions.

**Construction Steps**

1.) To open the construction window shown in Figure 2, click the **Window** menu and then click the **2-dim** option.

2.) Next we will show the Wingeom toolbar. The toolbar displays the tool that we can use to draw and manipulate geometric figures. To display the toolbar, click the** Btns** menu and then click** Toolbar**.

Figure 3 – The Wingeom window and its toolbar.

3.) The first step in our construction, we will draw the vertices of our triangle. To do this, right click three different locations on the drawing pad. Notice that Wingeom automatically names the points in alphabetical order.

4.) Next, to construct the sides of the triangle, select the **segments** option button in the toolbar then drag point *A *to point **B** to construct segment *AB*.

5.) Using the steps in 4, draw segments *AC* and *BC*.

6.) Next, we will draw the midpoint of *AB*. To do this, click the **Point** menu and then click on **Segment…** to display the new point dialog box.

Figure 4 – The new point dialog box.

7.) In the *relative to segment*, type *AB*, leave the coordinate to 1/2 and then click the **mark** button. Notice that point now lies on *AB*. This means that Wingeom should construct a segment halfway of *A**B*.

8.) To create a midpoint of *AC*, delete the text in the **relative to segment** text box and type *AC*. Then click the **mark** button. Notice that point *E* now lies on *AC*. Click the close button in the **new point** dialog box to finish.

9.) Draw segment *DE*. Refer to step 4.

10.) Let us see what happens if we drag the vertices of the triangle. To drag the vertices of the triangle, click the** drag vertices option button** on the toolbar, then drag the vertices of the triangle.

11.) Now, we will display the length of *DE* and *BC*. To display the length of *DE. To do this, *** **click the** Meas **menu to display the *measurements** dialog box*. Type *DE** *in text box of the *measurements dialog box *and then click the **ENTER **key.

Figure 5 – The measurements dialog box.

12.) Next type *BC* in the text box and then press the **ENTER** key. What can you observe about the lengths of segments *BC* and *DE*?

13.) Select the **drag vertices** option button on the toolbar and drag the vertices of the triangle. Is your observation still the same?

14.) Next, we will try to observe the relationship among the interior angles of two triangles – triangle **ABC **and triangle **ADE**. First we will display the measure of angle *ABC*. To do this, type **<ABC **in the text box of the measurements dialog box and press the ENTER key.

15.) Display the measures of the following angles using step 14: *ADE*, *AED* and *ACB*.

16.) What do you observe about the measures of the interior angles of triangle *ABC*?

17.) Close the measurements dialog box and drag the vertices of the triangle (refer to step 13). Are your observations still the same?

18.) Based on your observations, make a conjecture about the relationship of triangle *ABC* and its midline *DE*.

19.) Prove your conjectures.