Steiner Point Instructions using Geometer’s Sketchpad – 3 Points The shortest distance between 3 points uses a Steiner point. This point forms three 120 o angles in the interior of the points. The construction uses Torricelli’s method and involves basic geometry. By constructing a 60 o angle inscribed in a circle, the circle is divided into 120 o and 240 o sectors. Launch Sketchpad and open a new sketch. If the toolbox shown at the right is not visible, select Display – Show Toolbox. Use the Point Tool to create three points on the sketch. Next, label each point. Use the Selection Arrow Tool to select all three points. Note: When the selection arrow turns to a horizontal arrow over an object, clicking the mouse will select the object. When an object is selected, highlighted circle will surround the object. When all three points are selected, from the Display menu, choose Show Labels. Labels A, B, and C will now be placed on the points. You may click over the label and drag the label to a new position as necessary. You may also click and drag the points to a new location. Sketchpad Note: A common issue with sketchpad is having multiple objects selected. When selecting objects it is best to first click on the whitespace. Clicking the whitespace clears any object which may have been selected while creating other objects. If you forget to click the whitespace, choose Edit – Undo and follow the correct procedure. Figure 1 Torricelli’s Method 1. Create the vertices of an equilateral triangle using two points as a side of the equilateral triangle. We will construct two circles from points B and C with the radius of segment BC. The intersection will form an isosceles triangle. For our demonstration, we choose to use points B and C. To construct the triangle, click the whitespace to clear any selections. Select Point B followed by Point C. From the Construct menu, choose Circle by Center+Point. A circle will be drawn with B as the center passing through point C. Clear the selections by clicking the whitespace. Select Point C followed by Point B. Construct the circle using these two points. Next, find the intersection of the two circles on the exterior of the three points A, B, and C. Click the whitespace to clear any selections. Place the selection cursor over the intersection as shown in Figure 2. Click the intersection of the circles to create the point of intersection. For future reference, label this point. Select the intersection point and Display – Show Label. The point is now labeled as D. Figure 2
The circles with centers B and C are no longer required. Click the whitespace and then select the circles surrounding points B and C. Choose Display – Hide Circles. DO NOT delete the circles. The two circle should not be hidden with point D remaining on the screen. Steiner Point Instructions for Geometer’s Sketchpad Page 2 2. Circumscribe a circle around the equilateral triangle formed by points B, C, and D. The intersection of two angle bisectors of a triangle forms the circumcenter. Selecting 3 points allows for the construction of the angle bisector with the vertex angle being the middle point selected. Select points D, C, and B. From the Construct menu, choose Angle Bisector. Click the whitespace and select points D, B, Figure 3 and C. Construct the angle bisector. The intersection of the two angle bisectors is the circumcenter of the triangle formed by points B, C, and D. Click the whitespace and use the selection tool to find the intersection of the angle bisectors. Select the intersection and click it. A point should now be shown at the intersection of the angle bisectors. Construct the circle circumscribing the triangle created by points B, C, and D. Click the whitespace and select the point at the intersection of the angle bisectors. This point is the center of the circle. Next, select any of the vertices of the triangle such as point C. Choose Construct – Circle by Center+Point. Since the angle bisectors and center point of the circle are no longer needed, click the whitespace, select both angle bisectors and choose Display – Hide Objects. The construction should now be similar to the image at the Figure 4. 3. Construct the Steiner Point. The intersection of line segment AD and the circle is the Steiner Point. Click the whitespace to clear any selections. Select points A and D. Choose Construct –Segment. Find the intersection point by clicking the whitespace. Position the cursor of the intersection of Segment AD and the circle. Click the point of intersection. With the intersection point selected, choose Display – Show Label. The intersection will be labeled as point E. Figure 4 Figure 5 Point E is an internal junction point known as the Steiner Point. The circle, line segment AD and point D are no longer needed. Click the whitespace and select each of the objects. Choose Display – Hide Objects. The construction of Steiner Point E for points A, B, and D is now complete. Construct the Steiner Tree by creating segments AE, BE, and CE. Click the whitespace to clear all selections. Select points A and E. Choose Construct – Segment. Repeat the process of clearing the select and construct segments BE and CE. The complete Steiner Tree should appear similar to Figure 6. Figure 6
Steiner Point Instructions for Geometer’s Sketchpad Page 3 Further investigations using Geometer’s Sketchpad 1. Find the weight of the Steiner Tree. The weight of the Steiner Tree is the sum of the lengths of each edge of the network. First, measure the length of each edge and sum the edges. To measure an edge, clear the selection and select the segment by clicking in the middle of the segment. Click in the middle of segment AE and choose Measure – Length. The length of the segment should appear on the sketch. Repeat the process of measuring the length for segments BE and CE. The measurements may be moved to a different portion of the sketch by clicking and dragging. To calculate the total weight of the tree, click the whitespace to clear all selections. Select each of the three textboxes containing the measurements of the segments. Choose Measure – Calculate. From the Values drop down menu, choose one measurement at a time followed by the + sign to create a formula to add the measurements together. The formula window should be similar to: mAE + mBE + mCE . Click the OK button and the sum of the measurements should appear in the sketch. The total will update as the points are moved around the sketch. Figure 7 2. The measures of the angles between the edges at a Steiner Junction are to be 120 o . Measure the angles to confirm each is 120 o . Clear all selections by clicking the whitespace. To measure an angle, select the three vertices forming the angle. The vertex of the angle must be selected as the second point. Select points A, E, and B. Choose Measure – Angle. The mÐ AEB should be 120 o . Repeat the process to find the measures of Ð BEC and Ð AEC . Confirm the measure of each angle. Drag points A, B, or C. What happens to the angles? 3. Move point C so that the mÐABC > 120 o . What happens to the “Steiner Point”? Figure 8
Steiner Point Instructions for Geometer’s Sketchpad Page 4 3. Test other points to determine if there is a shorter network than the Steiner Tree. Use the Point Tool from the toolbox to add a point to the sketch. Choose the Selection Tool from the toolbox. With the point selected, choose Display – Show Label. The label on the point should be F. We will use F as a point to move around the sketch and measure the total weight. Measure the distance from F to each point A, B, and C. Click on the whitespace and select points F and A. From the Measure menu, select Distance. The distance from F to A should be shown on the sketch. Repeat the process to measure the distance from F to B and F to C. Calculate the total weight of the network A, B, C with a junction point at F. Click the whitespace to clear all selections. Click the three textboxes measuring the distances from point F. With the three textboxes highlighted, choose Measure – Calculate. Using the Value dropdown menu box to enter the distances, create the formula FA + FB + FC and choose OK. Click and drag the measurement textboxes to appropriate positions on the sketch. Compare the total weight of the tree using point F with the Steiner Point E. Figure 9
Steiner Point Instructions for Geometer’s Sketchpad Page 5 Steiner Point Instructions using Geometer’s Sketchpad – 4 Points Using a similar construction for two pairs of points may be used to find two Steiner Points. By connecting the two Steiner Points as junctions, a minimal spanning network may be constructed. These instructions assume the reader has an understanding of the Steiner Point Construction of 3 points discussed previously. Start a new sketch or from the File – Document Options menu, choose Add Page – Blank Page. Use the Point Tool to create 4 points on the sketch. Select each of the four points and label them A, B, C, and D similar to Figure 10. Figure 10 Using points A and B, construct an equilateral triangle with the point on the exterior of the quadrilateral ABCD. Label the point created for the equilateral triangle as E. (Labels may be renamed by double­clicking on the label.) Hide the circles with centers at A and B. Construct the angle bisectors to find the circumcenter and circumscribe the circle through vertices A, B, and E. Hide the angle bisectors and center of the circle. DO NOT hide the point labeled as vertex E or the circumscribed circle. Repeat the process using points C and D to create an external point F. The completed portion of this construction will appear similar to the Figure 11. Figure 11 Construct the line segment EF. Select points E and F. Choose Construct – Segment. Construct and label Steiner points G and H at the points where the segment EF intersects the two circles. Label the points G and H. Several objects are no longer needed. Hide circles ABE and CDF. Hide segment EF along with points E and F. The sketch should be similar to Figure 12. Construct the segments of the Steiner Tree (Figure 13). Figure 13
Figure 12 Steiner Point Instructions for Geometer’s Sketchpad Page 6 For further investigation: Note: If you wish to keep the previous construction intact, go to File – Document Options – Add Page – Duplicate and select the page with the 4 point Steiner Construction. A duplicate copy of the sketch will be added with a tab at the bottom of the page to select the previous sketch(es). 1. Measure all of the angles between the edges connecting the junction points. Is each angle 120 o ? 2. Measure the edges of the tree. Find the total weight of the Steiner tree. 3. Measure the lengths of AB, BC, and CD. Find the total weight of these three segments. Is it possible to arrange the points A, B, C, and D so that the Steiner Tree with interior junction points has a greater weight than spanning tree with no interior junction points? (See Figure 14 – The distance between the vertices can be measured without connecting the lines!) Figure 14
4. Add points I and J. Measure the distance from I to A and B, J to C and D and from I to J. Is it possible to position points I and J so that the total weight is less than the Steiner Tree? (See Figure 15) Figure 15