Centers of Triangles.MW



Chapter 3: Exploring Centers, Balance Points and Loci

Most high school geometry courses include the construction of at least four special points or “centers” of triangles. Two of these four are indeed centers of special circles related to the triangle: the Incenter is the center of the inscribed circle (incircle) and the Circumcenter is the center of the circumscribed circle (circumcircle). The other two special points are called the Orthocenter (the intersection of the three altitudes of the triangle), and the Centroid (the intersection of the three medians of the triangle).

The first remarkable property of each of these centers is that each one is the intersection point of three special lines or line segments. Why is this remarkable? Any two non-parallel lines will have only one point in common, but for three non-parallel lines to have only one point in common is very extraordinary. Try drawing three arbitrary, non-parallel lines on a sheet of paper. How many points of intersection do the three lines generate?

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Figure 3.1: Four Centers of Triangles

Activity 3.1

Construct each of the four triangle centers mentioned above using GSP. Figure 3.1 illustrates possible constructions. Construct the third special segment or line for each center and then vary each of the triangles to verify that the three segments or lines do appear to intersect at one point. Use Sketchpad’s ability to dynamically vary your constructions, together with the measurement capabilities, to come up with one conjecture or rationale for each of the trio of special segments intersecting in only one point. Such rationales are often overlooked in the geometry class and in Sketchpad workshops!

Activity 3.2

Create a separate GSP custom tool for each of the four centers. The givens for each custom tool should be the three vertices of the defining triangle. [Note: It is advisable to use LINES rather than SEGMENTS in all of the constructions in figure 3.1. You can hide the lines after creating the intersection point.] Save each custom tool as a separate sketch in your Tool Folder.

Investigating Properties of the Centroid of a Triangle

Open your Centroid sketch. The first investigation concerns the three medians of your triangle. Two of them were already used to construct the centroid. Display these first two by going to the Display menu and choosing Show All Hidden. The two medians and midpoints should appear on your triangle. Now construct the third median of this triangle.

The medians subdivide the triangle in some very special ways. The centroid subdivides the medians in a special way. To investigate these properties you will need to know how to use the Measure menu and the GSP Calculator.

You can measure the distance between two points by selecting both points and choosing Distance from the Measure menu. The distance measured will appear in your sketch window as a text object. This text object can be dragged around the screen, selected, hidden and displayed just like any other object. The measurement will change if you change the relative distance between the two points.

Use the Measure feature to find out how the centroid subdivides the medians.

You will probably want to calculate the ratio of two measurements. To do this, first select the two measurements then go to the Measure menu and choose Calculate from the bottom of the menu. A dialog box that looks like a calculator should appear (see Figure 3.2).

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Figure 3.2: The GSP Calculator Dialog Box

The thin rectangular box on the left side of the calculator labeled “Values” is a “pop-up” menu. Clicking on it should display the names of both of your measurements. [BG and GD in the example shown in Figure 3.2]. You should also see other items on this menu: New Parameter, the Greek letter π and e. You can select any item on this menu to be displayed in the calculator window by sliding down to it and releasing the mouse button. If you don’t like your selection click on the left arrow button (just above the OK button) and it will be removed from the calculation window. You can also use any of the calculator keys (numerals, parentheses, and operators) to build a mathematical expression in the calculate window.

• Build an expression for the ratio of your two measures. Select your first measure from the pop-up menu, then select the ÷ from the calculator keys, and finally select your second measurement from the pop-up Values menu. You should have an expression in the calculate window similar to the one shown in Figure 3.3. Click on the OK button to display this ratio in your sketch.

[Calculator Note: You could have inserted the measures into the calculator window by simply clicking on the measure in your sketch window once the calculator window is open, instead of using the Values pop-up menu.]

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Figure 3.3: Calculating the ratio of two measures in GSP

• Is this ratio the same for all three medians?

• Is it the same for any triangle?

• Does the ratio change if you change the length of your median?

• Write a conjecture concerning this ratio.

In order to investigate how the medians subdivide the triangle it is necessary to construct the polygon interiors of the small triangles formed by the medians. To do this, select all three vertices of one triangle and choose Triangle Interior from the Construct menu.

[Note: If Triangle Interior was not highlighted when you went to the Construct menu then you may not have selected all three points, or you had something else selected as well as the three points. Try making your selections again.]

The interior of the triangle should be filled with a colored cross-hatch pattern. This indicates that the interior is now selected. You could choose a different color from the Display menu at this point to change the color of the interior.

• Measure the area of this triangle by selecting its interior and choosing Area from the Measure menu.

• Construct and measure the interior of other triangles in your figure (including your original triangle).

• Calculate the ratios of the areas of some of your measured triangles using the GSP Calculator.

Change the shape and size of your original triangle. What happens to the ratios of the areas? Write some conjectures concerning the areas of triangles formed by the medians of a triangle. Share your conjectures with others in your class.

Balancing Triangles and Quadrilaterals

Cardboard triangles on tennis balls

You will work with a partner for this activity. Each pair of students will need a tennis ball and small paper cup to act as a fulcrum (see Figure 3.4). You will also need several different cardboard triangles. Work with your partner and try to balance your cardboard triangle on the tennis ball. The plane of your triangle should be horizontal. Mark the balance point with your pencil. Can you construct the balance point geometrically? (All you need is a ruler and pencil.)

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Figure 3.4: Tennis Ball Fulcrum

Finding the balance point of a triangle with GSP

If you came up with a construction for the balance point of a triangle, construct it with GSP. Can you use your Centroid tool to find the balance point?

Cardboard quadrilaterals on tennis balls

Do the same experiment with an irregular cardboard quadrilateral. Is it as easy to find the balance point of this figure as with the triangle? Explore with your partner ways to geometrically construct the balance point of your quadrilateral. Discuss your ideas with other pairs of students. Look at the different quadrilaterals and see if your ideas would work for each of them.

Finding the balance point of a quadrilateral with GSP

Activity 3.3

Find a way to use your Centroid tool for a triangle to construct the balance point of any convex quadrilateral in GSP.

Hint: Is a median a balance line of a triangle? If so, why? If you have two triangles that share a common side, what two points would a balance line for this shape have to pass through?

Demonstrate that any line passing through the balance point (center of gravity) of a convex quadrilateral will be a balance line for that quadrilateral.

Assignment 3.1

Extend your construction strategy for the balance point of a quadrilateral to find the balance point of any convex pentagon.

General Discussion

Share your approaches to the above explorations with the other members of the class.

What construction methods were used to find the balance points of the triangles?

What construction methods were used to find the balance points of the quadrilaterals?

Discuss the ideas that did not work. What assumptions were made? Which assumptions proved to be false? Why?

In what ways does the construction of the center of gravity (CG) of a triangular region generalize to the construction of the center of gravity of a quadrilateral region? In what ways does it not generalize?

Can the construction strategy for the center of gravity of a quadrilateral be generalized for any convex polygon? What about concave quadrilaterals? Will the construction still work?

On the CD accompanying these materials you will find a folder named Balancing a Quad. Inside this folder are various GSP sketches with custom tools to enhance this investigation and demonstrate possible solutions. The file Line Thro' CG of Quad simulates the balance line of a quadrilateral. The measurements and calculations show the relationship of the areas of the different regions formed by the balance line and their moments about the balance line. Change the position of the balance line by rotating it about the center of gravity. What happens to the areas and the moments?

Using coordinate geometry and algebra to find the center of gravity

The Geometer’s Sketchpad has a coordinate system that you can use to find the coordinates of any points in your sketch and also to plot points based on specific coordinates. We shall use this system to find out if it is possible to calculate the coordinates of the center of gravity of various polygons based on the coordinates of the vertices of the polygon. Start with a new sketch and create a triangle from 3 free points. Select all three points and go to the Measure menu. Choose Coordinates from this menu. Three pairs of coordinates should appear in your sketch along with the coordinate axes. How might we use these coordinates to find the balance point of the triangle? To answer this question, we need to make some assumptions about the physical situation that we started with. We assumed that our cardboard triangles were of uniform density and thickness, thus the area of the triangle could be used as a measure of the mass of the triangle. Let’s think about a simpler situation. Suppose we take one side of the triangle as a segment with two endpoints. Now assume that the endpoints are point masses of equal magnitude and that the segment is a uniform rod. Where would the balance point for the segment be located? The midpoint of the segment makes sense as a balance point in this “ideal” situation. How would you calculate the coordinates of the midpoint?

The midpoint will lie half way along the segment between the two endpoints. The x-coordinate of the midpoint will be half way between the x-coordinates of the endpoints and similarly the y-coordinate will be half way between the y-coordinates of the endpoints. You can use the GSP calculator to calculate these “half-way” coordinates. They will simply be the average of the x-coordinates of the two endpoints and the average of the y-coordinates of the two endpoints. So, if the two endpoints are labeled A and B, the x-coordinate of the midpoint will be (XA+XB)/2 and the y-coordinate will be (YA+YB)/2.

[GSP tip: You can obtain the x-value (abscissa) and y-value (ordinate) of each point by selecting the points and then choosing these options from the Measure menu. You can then use these measures in the GSP calculator to calculate the averages of the x-coordinates and of the y-coordinates of the two points. Remember to use parentheses in the calculator where they appear in the expressions.]

Once you have obtained the measures for the x and y-coordinates of the midpoint, select them (in that order) and use Plot As (x, y) from the Graph menu to plot the midpoint. Check that this plotted point really is the midpoint of your segment by using the Construct menu to construct the midpoint of the selected segment. The constructed midpoint should coincide with the plotted point. Move the vertices of your triangle (or endpoints of your segment) to check that the two points remain coincident. Notice that all of your measured and calculated coordinates change as you move the points. In Figure 3.5, point G is the constructed midpoint of segment AB.

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Figure 3.5: A Triangle and its Coordinates

You now have a new way to find the midpoint (balance point) of a segment using the coordinates of its endpoints. How could you extend this method to find the balance point (centroid) of the triangle? Let’s take another look at the physics of the situation. We could replace the segment AB and its 2 endpoints with a point mass at its midpoint, G. This point mass, however, would have to be twice the mass of either endpoint (why?). [Note: We are using a physical property of a rigid structure when we replace the structure by a point mass equal to the mass of the structure at the balance point of the structure.] We could now find the balance point between the segment (replaced by its midpoint G) and the opposite vertex, C. The balance point would have to lie somewhere on the segment GC (a median of the triangle ABC), but where? You might think it would be the midpoint of the median, GC but this would assume that the endpoints of the median were point masses of equal value. Our argument above should convince you that the point G (when replacing segment AB in the physical system of the triangle ABC) has mass twice that of point C (assuming a uniform triangle to start with). Have you ever tried to balance a see-saw with someone twice your weight ? Where on the see-saw should the heavier person sit (closer to or further away from the fulcrum)? If you are not sure about the answer to this question try a little experiment with a ruler and pencil. Rest the ruler across the pencil so that it balances (doesn’t touch the table the pencil is lying on). Now place a quarter (25 cents) close to one end of the ruler. Place 2 quarters on top of one another on the other side of the ruler so that the ruler again balances. Where did you have to place the 2 quarters in order to balance the one quarter? Use the data from this experiment to determine where the balance point should be for the points G and C on the median GC. Does this help explain why the centroid divides a median in the ratio 2:1?

In order to find the coordinates of the balance point between G and C we need to take into account the “weighted” values of each point. That is, that point G is twice point C in “value”. The total value of all points is 3 (3 times the value of point C). So, to find the “average” we need to add twice G to C and divide by 3. To find the x-coordinate of the balance point we calculate the following: (2XG+XC)/3. The y-coordinate will be (2YG+YC)/3. Verify, using the GSP calculator that these coordinates simplify to the following: (XA+XB+XC)/3 and (YA+YB+YC)/3. Plot these two measures as (x, y) in your sketch and verify that this plotted point is the same as the centroid of the triangle.

The above result seems to suggest that all we have to do to find the balance point of a set of points (the vertices of a polygon) is to take the average of the x-coordinates and the average of the y-coordinates, and these averaged coordinates will give the coordinates of the balance point of the polygon. So, for a quadrilateral with vertices ABCD, we would expect the balance point to be at the point {(XA+XB+XC+XD)/4), (YA+YB+YC+YD)/4}. Use your method for finding the balance point of any quadrilateral in GSP from your experiments with balancing cardboard quadrilaterals, then measure the coordinates of your quadrilateral, calculate the average coordinates and plot as (x, y). Does the plotted point always coincide with your constructed balance point? If not, can you explain why?

In Figure 3.6, the point F is the plotted point using the average coordinates of quadrilateral ABCE, the point cg(ABCE) is the balance point using the method of partitioning the quadrilateral into two different pairs of triangles (ABC, CEA and EAB, BCE), connecting the centroids of each pair of triangles with a line, and using the intersection of these two lines as the balance point of the quadrilateral). This method proved successful for our cardboard quadrilaterals (why is it a valid method?). Why does the average of the four vertices (point F) not coincide with this center of gravity? Experimenting with limiting situations may help explain the dilemma. Move point A very close to point B and observe what happens to points F and cg(ABCE). The quadrilateral gets very close to becoming a triangle and cg approaches the centroid of that triangle. Point F, however, moves closer to the “double point” at A and B (why?). If we were modeling the balance point of four point masses located at the vertices of our quadrilateral ABCE then point F would be the balance point (why?). When we balanced our cardboard quadrilaterals, however, we were balancing planar regions with assumed uniform density. The mass was assumed to be proportional to the area of the plane figure.

Assignment 3.2

Provide an explanation for why the two models (point masses at vertices and area of planar regions) give the same balance point for a triangle but not necessarily for any other polygon.

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Figure 3.6: Balance Points of Quadrilateral ABCE?

The Euler Segment and Centers of a Triangle

In order to explore relations among the different centers of triangles you need to construct all four centers on one triangle. Use your custom tools to construct the four centers on the same triangle and hide all of your construction lines to make it easier to observe and test any possible relations.

When you have all four centers on one triangle, drag a vertex of your triangle.

1. Which centers appear to be co-linear? Are these centers co-linear for any triangle?

2. Are there any triangles where all four centers are co-linear?

You are going to use the Animation and Trace Locus features of GSP to explore what is known as the Euler Segment (a segment connecting the Circumcenter and Orthocenter of a triangle), and the loci formed by the different centers as a vertex of the triangle is animated along a segment.

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Figure 3.7: The Euler Segment

1. Connect the Orthocenter (OC) and the Circumcenter (CC) of your triangle with a segment (see Figure 3.7). Make each center a different color.

2. Draw a segment across the top of your sketch. Select this segment and the top vertex of your triangle.

3. Go to the Edit menu and choose Merge Point To Segment. Your top vertex should move onto the segment.

4. With your merged vertex still selected, go back to the Edit menu and select Action Button-Animation. An animation properties window should appear. Click the O.K. button to accept the default properties. An Animate Point button should appear on your sketch (see Figure 3.7).

5. Double click on the Animate Point button. What happens? Click back on the button to stop the animation.

6. Select point OC. Under the Display menu choose Trace Locus.

7. Start the animation again. What shape is traced out?

8. Change the relationship between the triangle and the line segment. (move one end-point of the line segment). What happens to the traced locus?

9. Choose any of the other centers and trace its locus.

You can construct the Locus of the Orthocenter as a vertex of the triangle moves on your free segment. To do so, select the point OC, and the vertex of your triangle that is on the free segment in that order. Go to the Construct menu and select Locus. The locus should appear as a continuous curve. Change your triangle. Change your free segment. What happens to the locus of the orthocenter? What kind of a function might generate this curve? (What kind of a curve does it look like?) Discuss your findings and conjectures with the rest of the class.

The Power Plant Problem: Finding Another Center of a Triangle

A power plant is to be constructed to serve three cities. The company wants to minimize the amount of high-voltage cable used to deliver the electricity to each of the cities. Where should the power plant be built? (Assume that the three cities do not lie in a straight line, and that high-voltage power lines can be run in straight lines from the power plant to each city.)

Activity 3.4

Use the vertices of a triangle in GSP as the three cities. Create a free point in your triangle and measure the distance from this point to each vertex. Sum these three distances. Use this sketch to help solve the power plant problem. Try and find a way to construct the point where the power plant should be, given any triangle. [Hint: You might also measure the angles subtended by the Power Plant and each pair of vertices (angles APB, APC, BPC in triangle ABC). Is there anything special about these angles when the sum of the distances is a minimum?]

In the particular situation shown in Figure 3.9, the mayor of city A claims that the power plant should be built at the center of city A. Is the mayor correct? Why or why not?

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Figure 3.9

There is a physical way to find this center using cardboard or other flat, rigid surface. Draw a triangle on your cardboard surface. Make holes at each vertex of your triangle. Knot three smooth pieces of string together and thread one piece through each vertex. Tie equal weights to the ends of each string (or one weight to all three strings?) and let the weights pull the strings taught. The knot joining the three strings should come to rest at a point that gives the shortest distance from all three vertices. Why?

Ceva’s Theorem

The exploration of triangle centers began with constructing special trios of lines associated with the triangle. Each of these trios of lines had a single point of concurrency (intersection). It is possible to construct a trio of concurrent segments (called Cevians after the mathematician, Ceva) in a triangle starting with free points on two of the sides of the triangle. Connect each free point to its opposite vertex. Construct a line through the intersection of these two segments and the third vertex. Where this line intersects the third side of the triangle creates the third segment in the trio. In Figure 3.10, D and E are the free points. BG is the third segment.

Figure 3.10: Constructing three Cevians, CD, AE and BG in a Triangle

Assignment 3.3

Investigate the product of the ratios of the subsections of each side of the triangle: (BD/DA)*(AG/GC)*(CE/EB) measuring clockwise around the triangle from B. Move the free points D and E. Change your triangle. Form a statement for Ceva’s Theorem.

Reflections and Extended Reading

In the above explorations we used at least three different technologies (cardboard surfaces, paper folding and GSP). Reflect on how the investigations using one technology were modeled by another technology. Did these interplays among the technologies help strengthen your understanding of the mathematical constructs? What are the implications of this interplay for teaching with technology?

The above explorations only scratch the surface of the rich geometry to be found in triangles and their associated centers and circles. Douglas Hofstadter (1998) has written an intriguing paper describing his own explorations of these “geometric gems” in which he uncovers new relations among our four original centers and many others. I encourage you to follow Doug’s journey in his paper, Discovery and Dissection of a Geometric Gem by constructing the special centers and circles with GSP. In this way you will be participating in a piece of original mathematics! You can also find many more GSP explorations of Centers of Triangles on Dr. Jim Wilson’s web page: .

References

Hofstadter, D. (1998). Discovery and Dissection of a Geometric Gem. In J. King and D. Schattschneider (Eds.) Geometry Turned On: Dynamic software in learning, teaching, and research. MAA Notes 41.

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