Morley s Theorem and Triangle within Triangle Lesson Plan

Morley s Theorem and Triangle within Triangle Lesson Plan By: Douglas A. Ruby Date: 10/8/2003 Class: Geometry Grades: 9/10 INSTRUCTIONAL OBJECTIVES: This is a two part lesson designed to be given over a several day time span. At the end of the first part of the lesson, the student will: 1. Understand new features of GSP 4.0.3 including: a. How to construct measured angles b. How to hide parts of a construction c. How to drop points onto intersections of lines d. How to use the text tool to assign arbitrary labels to points 2. Have constructed a triangle with rays that form angle trisectors from each vertex 3. Verified that the rays constructed in #2 remain angle trisectors of the vertex angles when the triangle is manipulated. 4. Discovered the geometric relationship formed when segments are drawn between the points of intersection of these trisectors. 5. Be able to state a theorem based on observation of the relationships (induction). 6. Research the background on this theorem (Morley s Theorem) and the methods used to prove it using Web resources with proper bibliographic references. 7. Written a summary of the difference between a logical deductive (Euclidean) proof and an analytic proof. At the end of the second part of the lesson, the student will: 1. Understand new features of GSP 4.0.3 including: a. The meaning of dilation to create points on the sides of a triangle b. How measure the area of a triangle 2. Created a triangle within a triangle using dilated points 3. Measured the areas of the outer and inner triangle 4. Explored the relationship between the area of the outer and inner triangles. 5. Explored and researched possible reasons for the relationship discovered in #4. Page 1

Relevant Massachusetts Curriculum Framework 10.G.1 Identify figures using properties of sides, angles, and diagonals. Identify the figures type(s) of symmetry. 10.G.2 Draw congruent and similar figures using a compass, straightedge, protractor, and other tools such as computer software. Make conjectures about methods of construction. Justify the conjectures by logical arguments. 10.G.3 Recognize and solve problems involving angles formed by transversals of coplanar lines. Identify and determine the measure of central and inscribed angles and their associated minor and major arcs. Recognize and solve problems associated with radii, chords, and arcs within or on the same circle. 10.G.4 Apply congruence and similarity correspondences (e.g., ABC XYZ) and properties of the figures to find missing parts of geometric figures, and provide logical justification. 10.G.5 Solve simple triangle problems using the triangle angle sum property and/or the Pythagorean Theorem. 10.G.6 Using rectangular coordinates, calculate midpoints of segments, slopes of lines and segments, and distances between two points, and apply the results to the solutions of problems. 10.G.7 Draw the results, and interpret transformations on figures in the coordinate plane, e.g., translations, reflections, rotations, scale factors, and the results of successive transformations. Apply transformations to the solutions of problems. INTENDED AUDIENCE This lesson is targeted at a 9/10 th grade honors geometry audience. These students already know how to use Windows 98/Me/XP based computers and are fluent with use of keyboard, mouse movement, left and right mouse click, and normal conventions and terms such as <esc>, <tab>, double-click, etc. Students have already taken Algebra (either in Middle school or high school) and have already been introduced in prior classes to the notions and conventions of formal proof, logic, and the basics of Theorems, postulates, definitions, and construction. Page 2

CLASS ACTIVITIES Part 1 - Trisecting Angles within a Triangle In our previous lesson, we learned how to use Geometer s Sketch Pad Version 4.0.3. In this lesson, we will build on that knowledge. Our purpose in this lesson is to create a construction that cannot easily be created using the traditional Euclidean means of a compass and straightedge and then to observe and discover the properties of this construction. 1. Create a new workspace and save it as A:trisect.gsp We begin by starting Geometer s Sketchpad and using File>Save As to save the blank workspace as A:trisect.gsp. 2. Create Triangle ABC Create a triangle ABC using the Point Tool, Text Tool, and Construct>Segments menu as you did in previous lessons. 3. Construct Ray AB to extend side AB Use the Selection Tool. Select Point A (first) and then point B. Construct Ray AB by using the Construct>Ray menu. If you select the point B before point A, you will end up creating Ray BA rather than Ray AB. How are these different? A ray is a set of points that begins at an endpoint A and continue infinitely through another point B. The first point we select is the endpoint 4. Construct Ray BC and Ray CA Now use the Selection Tool again and construct rays that extend the other two sides. Your triangle should now look like this: Make sure your triangle looks like this before you turn the page! Page 3

5. Measure angle BAC and calculate 1/3 of its value Using the Point tool to select points B, A, and C and then use the Measure>Angle menu to measure BAC. What is this angles measure? varies Now make sure that you have highlighted the measure of BAC. This should look something like this on your screen. Use the calculator (you did this in the previous lesson) to calculate one third of the measure of BAC. What is this value? varies 6. Trisect BAC Use the Select Tool to select Point A and then the Transform>Mark Center menu to mark point A as the center of rotation. You should see a little animation on your screen over point A. Now select the calculation we did in the prior step (1/3 of m BAC) and make sure it is highlighted in your workspace. It will look like this. (Note: actual numbers will vary from triangle to triangle) With this measurement highlighted, use the Transform>Mark Angle menu. You will see a different animation around the measurement box. Finally, select the ray AB created in step 3 and use the Transform>Rotate menu. You should see a new ray that begins at point A and extends through the opposite side of your triangle. It will look like this: If you do a Transform>Rotate a second time, you should see a third ray extending from point A. What have we just done? We just trisected the angle BAC How could you prove this? We could measure each angle that terminates at point A and verify that they are equal and that the sum of their measures is equal to BAC. Stop here and make sure you have successfully trisected BAC before you proceed. Page 4

Your triangle should now look something like this (though angles will vary): 7. Trisect angles ABC and BCA Now, we will repeat the process of trisecting an angle as we did for points will be points B and C respectively. Remember you need to: BAC, except the center a. Measure the angle b. Measure the angle 3 c. Mark the center of rotation d. Mark the angle of rotation e. Rotate the ray whose end point is the center of rotation (by the marked angle) f. Rotate the ray just created by the marked angle again. Each of the vertex angles of your triangle has now been fully trisected. Your construction should look like this: Page 5

8. Fill in the measure of the three interior vertex angles of your triangle below: m m m BAC = varies BCA = varies ABC = varies 9. Clean up the rays We now are going to make our workspace look better. We want to retain the trisected angles within the triangle, but hide the underlying rays that created them. To do this, we will use the Point Tool and drop points on the intersections of the rays and each of the sides of the triangle. To create a point at the intersection of two lines, move the point tool until it is over the two lines. Once both lines are highlighted, click the point tool. You should have just dropped a point on the intersection. If both lines were not highlighted, the point will be off center from the intersection. This will look like this: Once all of the intersections have points on them, we want to create line segments between each vertex and the two points on the opposite side of the triangle from that vertex. We do this by highlighting the vertex and an opposite point and then using the Construct>Segment menu. Now, we can hide the construction rays that we used to trisect the vertex angles. We do this by deselecting everything in our construction (click the Point Tool in white space) and then selecting all 9 rays used in our construction. Once all 9 rays are highlighted, use the Display>Hide Ray menu and the rays will disappear. Your construction will now look like this: If you missed creating a segment, you will miss one of the lines in the triangle above. If so, Display>Undo Hide Rays to cause them to be visible again. Stop here and make sure you have successfully created this triangle before you proceed. Page 6

10. Test your construction If you have created your construction properly, you should be able to click and drag a vertex of the triangle to test it. If you do this, what should happen? The shape of ABC will change, but the rays extending from each vertex should continue to be trisectors of the vertex angle. Try dragging your triangle around. Check the angle measurements and the interior segments. Do they behave the way you expect them to behave? What happens to your triangle as it changes shape? The shape of ABCchanges as we drag its vertices around. However, the rays extending from each vertex continue to trisect the interior angles. 11. Construct an interior triangle The interior of your triangle should look something like this: Using the diagram above, pencil, and an eraser, try creating a triangle between any three intersection points of the angle trisectors. An example might look like this: Does the triangle you just created have an interesting shape? What kind of triangle did you create? Answers should vary. Students might observe isoceles, right, obtuse, acute, or equilateral triangle. Erase the triangle you just created on paper and try a different set of intersections. Once you have created your interesting triangle on paper and believe you have discovered something you want to commit to your construction, you will construct that triangle within Geometer s Sketchpad. Write a conjecture about what kind of triangle you believe you will be creating: _We want the students to find the equilateral triangle, but they may not. 12. Create inner triangle DEF using GSP. Now, drop three points on the intersection points you selected above. Then with the three points highlighted, use the Construct>Segments menu to create the inner triangle. Use the text tool to label the three points D, E, and F. If you were to use the triangle above, these labels would look like this: (Note: this is not necessarily the correct triangle to use) Stop here and make sure you have successfully created DEF before you proceed. Page 7

13. Test your DEF construction You should now be able to test DEF created on the previous page. Based on the conjecture you assumed about DEF, you should expect certain characteristics about DEF. Using Geomters sketch pad and the Measure>Angle and Measure>Segment menu, measure each angle and segment of DEF. Write down the measurements for the angles and sides of DEF below: m DEF = varies DE = varies m EFD = varies EF = varies m FDE = varies FD = varies Based on these measurements, was your conjecture about the nature of DEF correct? Yes or no If yes, why? If no, why not? Students who chose equilateral, should find that the sides of the triangle have the same measure and that the angles are 60 o. If a student chose right triangle, they should see an angle with 90 o. If isosceles, at least two sides should have the same measure._ If you believe that you have proven your conjecture by measurement, we have one more test. We want you to grab one of the vertices of the original ABC and drag it around so that the shape of the outer triangle changes. What happens to the inner triangle? Should remain same kind. What happened to your measurements? answers vary Do you measurements still support your conjecture? answers vary If the answer to the last question is YES, you may turn the page. If the answer to the last question is NO, proceed to step 14. 14. Fix your construction. Is you are here, then DEF did not support your conjecture or your initial conjecture was false. If so, we need to delete the triangle by selecting points D, E, and F and hitting the <del> key. When the three points of intersection are deleted, the segments connecting them will be deleted too. Now using three different points of intersection, drop points on them and label them D, E, and F. Create a new triangle. The three points may have new names like G, H, and I. You can change these labels by using the point tool after the labeling is done. Select the point whose label you want to change, right click, and use the properties menu item. This will allow you to change the label of the point with a popup that looks like this: Once you have created a new DEF, go to step 13 and retest your construction and new conjecture. Do not proceed until you have a DEF that supports your conjecture Page 8

15. Congratulations, you have gotten this far! If you are on this page, you should have found out that the special inner triangle is an equilateral triangle regardless of the size and shape of the outer triangle. You should now have a construction that looks like this (some measurements will vary): In particular, DE = EF= FD and the m DEF = m EFD = m FDE =60 o. You should be able to grab a vertex of your ABC, change its shape, and the inner DEF should remain an equilateral triangle. Now that you have created this construction, it is time to save it. Save your workspace to your floppy disk using the File>Save menu. 16. Write your conjecture down We now want you to write down your conjecture in the form of a theorem. Based on your observations, lets answer the following questions: What did we do to ABC to create the three points used in DEF? We trisected each interior angle of the triangle using rays whose angle was equal to 1/3 of the interior angle each ray emanated from. Which specific points of intersection did you have to use to create the equilateral triangle DEF? We had to use the intersections of the adhjacent trisectors. That is, the intersection points nearest to the sides of the original triangle. Now, write your statement of this theorem: _The intersection points of adjacent trisectors of any triangle form the vertices of an equilateral triangle. Do not proceed until you have written your theorem Page 9

17. Researching your theorem I am certain it is no great surprise, but the fact that the intersection of the angle trisectors of any triangle forms an equilateral triangle is not a new discovery. In fact, this property of triangles is known as Morley s Theorem. Morley s theorem has a number of proofs, but most of them are different from the Euclidean proofs that we have used so far in this class. We will use Web resources in order to discover some facts about Morley s Theorem and better understand the nature of its proof. Using a combination of Google Search from Google (http://www.google.com/) and resources outlined in the references at the end of this paper, research answers to the following questions: 1. Who was Frank Morley? Where, when, and what did he do? 2. The capability to construct a geometric figure using compass and straightedge alone (not including a protractor) is the basis for most traditional Euclidean constructions and proofs. In our construction, we used the 1/3 of measure of each vertex angle to create the rays that trisected the vertex angles. This is equivalent to using a protractor. Is it possible to trisect an angle with compass and straightedge alone? Provide some history on this topic. 3. Read one of the proofs of Morley s Theorem from your references. What do you notice is different about these proofs from the two-column deductive logic proofs you have been using so far? 4. Find a definition of the following two terms: a. Deductive Proof b. Analytic Proof Once you have done that, see which form of proof fits any of the proofs you have seen for Morley s Theorem. What kind of proof is used for Morley s Theorem? Why do you think that form of proof was used? Write your answer to these questions and any other observations you have from your resources. Make sure you document your references in a way similar to the references at the end of this document. References may be checked for originality (i.e. plagiarism) and correctness. Page 10

Part 2 - Creating a Triangle Within a Triangle In our previous lesson, we learned how to use Geometer s Sketch Pad Version 4.0.3. In this lesson, we will build on that knowledge. Our purpose in this lesson is to create a construction that cannot easily be created using the traditional Euclidean means of a compass and straightedge and then to observe and discover the properties of this construction. 1. Create a new workspace and save it as A:triwithin.gsp We begin by starting Geometer s Sketchpad and using File>Save As to save the blank workspace as A:triwithin.gsp. 2. Create Triangle ABC Create a triangle ABC using the Point Tool, Text Tool, and Construct>Segments menu as you did in previous lessons 3. Create 1/3 dilation points on each side of the triangle. Now we will do something new. We will dilate vertex points (i.e. slide them) up the sides of the triangle by a factor based on the length of each side. Mark point A as a center by selecting point A with the Point Tool and then using the Transform>Mark Center menu. Point A will have an animation around it for a few seconds. Now, we will dilate point B up from point A by a scale factor of 1/3. To do this, you will select point B using the Point Tool and then use the Transform>Dilate menu. You should see the following: Change the ratio described in the popup window from 1.0 2.0 to 1. 0. Then select Dilate. 3. 0 Page 11

Once you have done this, use the Text Tool to label the new point. You should now see the following: In your own words, describe what we have done: Now, mark point B as a center and dilate point C by a scale factor of 1/3. Then, mark point C as a center and dilate point A by a scale factor of 1/3. 4. Connect the dots Now construct segments from each vertex to the 1/3 dilation points you just created. Your triangle should now look like this: B C' B' 5. Construct interior intersections. A A' C Now we will drop points on the three interior intersections of the lines we just created. Label these points D, E, and F. Page 12

6. Create Interiors In our first lesson, we created triangle interiors. We want to do this again. Select the outer triangle ABC by highlighting points A, B, and C and then using the Construct>Triangle Interior menu. Next create a triangle interior for DEF. Once this is done, you can change the colors of the interiors by highlighting the interior with a left-click of the Point Tool and using the Color popup menu after doing a right-click. This menu looks like: 7. Measure Area of Triangles Once both triangle interiors are created, we want to measure the area of each triangle interior. Do this by selecting the triangle interior and then using the Measure>Area menu. Write the area of each triangle in below: m ABC = Varies m DEF = Varies Using the Measure>Calculate menu, find the ratio of the larger triangle to the smaller triangle. What is this ratio? The ratio should be 7 to 1 Do not proceed until you have completed measuring the areas and comparing them. Page 13

8. Write a conjecture We now want you to write a conjecture regarding the ratio of the area of the outer triangle to the inner triangle. Please answer the following questions first: What did we do to create the 3 points on the side of ABC named A`, B`, and C`? We dilated slid ) the points 1/3 of the distance up the side of the triangle. This distance is equal to the dilation factor used in the transformation. How did we create the inner triangle? The inner triangle is created by marking the intersections of the three segments we created between the vertices of the triangle and the dilated points. Write your conjecture: _ The inner triangle( DEF) created by dilation of intersecting segments from the vertices of an arbitrary triangle ( ABC) has an area that is constant with respect to the area of its parent triangle based on the dilation factor used. 9. Test your conjecture How might you test to see if your conjecture is correct? We will move the triangle around and see if the ratio of the outer to inner triangle is still 7 to 1. Try dragging a vertex of ABC around to see if the ratio of the two areas changes. What happens? The ratio should stay the same Is your conjecture true? YES or NO If the answer to the last question is NO, go back to step 8, modify your conjecture, and retest it. If the answer to your last question is YES, continue by turning the page. Do not continue until you have successfully tested your conjecture Page 14

10. Triangle Properties You now should have a triangle and area calculations looking like this: Clearly, the ratio of the area of the outer triangle to the area of the inner triangle stays constant at 7.00 to 1.00 regardless of changes in the shapes of the triangles. Now we want to test other properties of these two triangles and try to find out why this ratio exists. Try to see if you can answer the following questions: a. Is triangle ABC similar to DEF? No it is not b. Make sure you test this assumption and do not just eyeball the two triangles. How do you do this? Measure the three vertex angles of ABC. Then measure the three vertex angles of DEF. You will see that they are not equal, thus the triangles cannot be s,milar. Try constructing a triangle based on the following different dilation factors. What is the ratio of the outer triangle to the inner triangle? Dilation factor Area of ABC Area of DEF ABC DEF 1/2 14.08 cm 2 0.0 cm 2 No solution = 1/3 21.16 cm 2 3.02 cm 2 7 1/4 12.90 cm 2 3.97 cm 2 3.25 1/5 12.90 cm 2 5.53 cm 2 2.33 What kind of relationship can we deduce from this data? In each case, the relationship between the area of the inner triangle is a constant. However, the constant varies based on the dilation factor. We do not yet understand this relationship. Save your workspace to your floppy disk using the File>Save menu. Do not continue until you have saved your work and completed the table above. Page 15

11. View the triangles another way. After creating an inner triangle whose dilations are 1/5 of the distance from the center to the opposite vertex, you should have something that looks like this: So far, we have been measuring the areas of the inner triangle DEF. However, now I would like you to measure the areas of ABC, AB C, and A BC. What do you find these to be? Each of these triangle have the same area. What is the ratio of the parent ABC to one of these smaller triangles? _ That area is equal to the 1/5 of the area of the total triangle. 1/5 is the dilation factor we used to create the points A, B, and C What is the area measure of AB D, BEC, and A CF? These triangles are also equal. The area is 1/105 th of the area of the original triangle AB D Triangle Area of ABC Area of ABC ABC 15.83 cm 2 3.17 cm 2 5 AB C 15.83 cm 2 3.17 cm 2 5 A BC 15.83 cm 2 3.17 cm 2 5 AB D 12.90 cm 2 0.15 cm 2 105 BEC 12.90 cm 2 0.15 cm 2 105 A CF 12.90 cm 2 0.15 cm 2 105 Using these various inner triangles, how do we calculate the area of DEF based on the inner triangles? m DEF = m ABC (3 m ABC ) ( 3 m AB D) What is the area of ABC, AB C, and A BC relative to the area of ABC and the dilation factor (DF)? m ABC = (DF)) m ABC What is the area of the small triangles AB D, BEC, and A CF relative to the area of the outer triangle ABC? m AB D = (1/21) DF m ABC Page 16

Do these same relationships hold true for dilation factors other than 1/5? The measure of ABC is always DF m ABC, however, the measure of the three smaller triangles varies by a factor of 21 that is not the same for every size. We have not yet discovered this completely. Can you write an equation that would calculate the size of ANY inner triangle ( DEF) based on the dilation factor (DF) used? m DEF = m ABC (3 DF m ABC) -?????? Don t know yet. We have now completed our investigation of the Triangle within a Triangle. Can you summarize what we learned? We learned how to dilate points based on a factor of the length of a side of a triangle. We learned how to measure the areas of triangles. We also learned that the measure of the interior triangle created within the outer triangle is a constant based on the dilation factor. Further investigation allowed us to learn that three overlapping triangles are created, each of which is exactly equal in area to the dilation factor (DF) times the area of the original triangle.however, we did not figure out the formula for the overlap of each of these three outer triangles, so that we do not yet have a complete formula for computing the area of the inner triangle based on the outer triangle. Page 17

Attachment A Sample Constructions for the two exercises in this lesson plan. m BAC = 53.24 m BAC = 17.75 3 m ABC = 50.47 C m ABC 3 = 16.82 m ACB = 76.29 m ACB 3 = 25.43 D E m DE = 1.08 cm m DF = 1.08 cm F m FE = 1.08 cm A B Morley's Theorem states that the intersection of the adjacent angle trisectors of a triangle form the vertices of an equilateral triangle. Area ABC = 21.16 cm 2 Area DEF = 3.02 cm 2 Area ABC Area DEF = 7.00 m DEF = 94.46 B m CBA = 92.70 m EDF = 37.75 m BCA = 36.46 E C' m BAC = 50.84 m DFE = 47.79 B' D F A A' C Discussion: Note, Based on the fact that we are dilating points by a 1/3 factor, the triangle within is always 1/7th of the area of the extyernal triangle. There clearly is a relationship between the dilation factor and the area. We will try to examine this analytically. Page 18

Attachment B References Bogomolny, A., Morley s Miracle, retrieved October, 6, 2003, from http://www.cut-theknot.org/triangle/morley/index.shtml St. Andrews University. The MacTutor history of mathematics. Retrieved October 6, 2003 from http://www-gap.dcs.st-and.ac.uk/~history Weisstein, E., Eric Weisstein s World of Mathematics; Morley s Theorem, retrieved on October 6, 2003 from http://mathworld.wolfram.com/morleystheorem.html Page 19

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