Stanley Wong's CU

Heron, Pythagoras, Euler, Ptolemy and Ceva:
Some Contributions to Euclidean Geometry

Stanley Wong

The Academic Setting

This unit will be used at Del Norte High School in Albuquerque, New Mexico. It is designed with my Honors Geometry class in mind. Students can be freshman or sophomores. The freshman are recommended by their eighth grade Algebra I teachers if they have exhibited a strong aptitude for mathematics. Math teachers from our feeder schools (Cleveland and McKinley Middle Schools) have been briefed on these recommendations. The sophomores are those who have successfully completed Honors Algebra I as freshman or who have shown great promise in regular Algebra I and who have been recommended by their teacher.

           Students in Honors Geometry will study, in greater depth, the concepts, techniques, and theory of the regular geometry course. Both acceleration and enrichment are integral components of the curriculum. This is the second course in the Honors/Advanced Placement Program in Mathematics and students will earn a weighted grade in this course. (Albuquerque Public Schools 16.0)

            It has been said that Del Norte High School reflects the ethnic make-up of the state of New Mexico more than any other high school statewide. It has a student population of about 1650 that is approximately 44% Hispanic, 38% Anglo, 8% Native American, 6% African-American, 2% Asian, and 2% other. About 18% of the students receive free or reduced lunch. The socioeconomic make-up of Del Norte is very diverse. Since there are many single family homes with families that have traditional ties to Del Norte, we have a generally stable population. However, the many apartments in the area also encourage a somewhat transient population.

Context and Background

Rationale

Students need to have a sense of history, of how and why they are living where they are. In social studies, we learn, among other things, how the United States became a country. We see how and why we have certain rules and regulations that help us to be able to have life, liberty and to pursue happiness. The founders of our nation have set forth principles of government so that we can have individual freedoms while not impeding the freedom of others. Understanding the history gives us a greater appreciation of what we have and helps us use it to the fullest. When we study George Washington, Benjamin Franklin, Thomas Jefferson, and other patriots of our country, we get a greater understanding of the importance of what we have in this country. We have the principles of what they stood for but historians also write about their lives in order to better understand those principles. We gain greater insight when we see why they stood for what they did. When we put ourselves in their shoes, we can better identify why they did what they did. This is also true in mathematics.

            When students learn mathematics in school, they may be fascinated by what it offers. They may be able to understand a theorem that leads to a formula. They may be able to use that formula to solve a practical problem. But who proposed that formula? When was it first used? Why was it discovered at that time? Would the world be different if some mathematical formula was discovered say 1000 years before or 1000 years after when it was discovered? Were formulas discovered out of necessity or did they happen randomly? What if the Washingtons, Franklins, and Jeffersons were born earlier or later? Would there be a United States of America.? I do not propose to answer these questions. I just want mathematics to be appreciated not only in the use of the theorems themselves but in the history of their development.

            When I was in high school, I had an English teacher who would tell us where certain words came from. It was very fascinating to me. It made me much more interested in a subject that I had great difficulty with. The derivation and the history of words gave the words more meaning. The derivation and history of mathematical theorems and formulas also lend greater meaning to their mathematical value. To know the mathematicians who are associated with mathematical discoveries also gives insight to the times when they were alive.

            Since the course in Honors Geometry is mainly Euclidean, I will look at the discoveries concerning more traditional aspects of geometry. I would like to look at five mathematicians and their discoveries.     This will give students a greater appreciation and enrichment of the geometry that they are studying. Heron’s Formula uses just the sides of a triangle to find the area. This can also be used to prove the Pythagorean Theorem, the second discovery that I will look at. The study of geometry is enhanced by geometric construction using only a writing utensil, a compass, and a straightedge. Euler’s Line and his Nine-Point Circle can be constructed though the proofs may be quite advanced for elementary geometry students. After students master similar triangles and their properties, they can prove Ptolemy’s Theorem and Ceva’s Theorem. Ptolemy’s Theorem also implies the Pythagorean Theorem. It is hoped that students will also see that the geometry that has been studied throughout the ages is intertwined.

            This unit would not necessarily be covered consecutively nor in one block of time. It should be used when appropriate as a particular topic is covered during the geometry curriculum. It is my intent in the Honors Geometry curriculum that each student presents a mathematical topic to the class. This unit will also give students some ideas for this oral presentation.

Background - Heron

It is well known that the formula for the area of a triangle is Area = one-half times the base times the height. However, in practical application, the three sides of a triangle are usually known and unless it is a right triangle, the basic formula cannot be used directly since an altitude must be found. It would seem that since a triangle with the lengths of its three sides given is uniquely determined that there should be a way to find its area using just the lengths of the three sides. A person thinking practically would see it this way. Heron of Alexandria lived sometime around AD 75. His interest in mathematics tended to the practical rather than the theoretical, and many of his writings dealt with such useful applications as mechanics, engineering, and measurement. He explained in his Dioptra how to dig tunnels through mountains and how to measure the amount of water flowing from a spring. How can we find the area of a triangle directly without needing to find the height? Heron’s formula showed this practicality by only using the sides of a triangle to find its area. If a, b and c are the lengths of the three sides of a triangle and s is the so called semiperimeter of the triangle (s= ½(a+b+c)   ), then the area of the triangle A = .

Heron’s proof of this theorem is a wondrous piece of abstract geometric reasoning (Dunham, Journey 118). His proof uses only simple ingredients from plane geometry. Yet Heron displayed an astonishing geometric virtuosity in compiling these elementary pieces into a remarkably rich and elegant proof that boasts one of the best surprise findings in mathematics. Readers of Heron’s proof can be within a few lines of the end and still have no idea how the matter will be resolved (Dunham, Journey 119). To look at Heron's proof, see pages 119-127 in Journey Through Genius, The Great Theorems of Mathematics by William Dunham.    Though this theorem had been known for centuries, Heron’s treatise on this was lost until 1894. It should be noted that historians have come up with manuscripts that credit this formula to Archimedes though no Archimedean writings have been found to support this claim. When I first saw this formula as a student teacher, it was named Hero’s formula perhaps because of the translation from the Greek but it is one that has always fascinated me because of its simplicity. There is another proof of the theorem using trigonometry which is presented in many pre-calculus textbooks. The proof using trigonometry is also a surprising proof and the mathematics used includes the Law of Sines and the Law of Cosines with some nifty substitutions.

            In order to use Heron’s formula, one only needs to be able to add, subtract, multiply, divide, and take a square root. A calculator is not really needed if one knows how to use the square root algorithm. Modern day students are not usually taught this algorithm but students who are in Honors Geometry can be taught this quickly and it lends a greater understanding to square roots and its relationship to the square of a binomial.

            Heron’s formula has many practical uses. If one wants to find the area of a triangular field, one only needs to use its triangular dimensions. It may not be practical to find any of the altitudes so Heron’s formula makes it easy. It also can be used to prove the most famous theorem in elementary geometry - the Pythagorean Theorem.

            One added note: Since a quadrilateral can be looked at as two triangles, it is possible that the area of a quadrilateral would have a formula using just its four sides similar to a triangle using just its three sides as Heron stated. The Hindu mathematician Brahmagupta discovered just that around AD 620. Brahmagupta’s generalization of Heron’s Formula, which is true for a quadrilateral inscribed in a circle (note that Ptolemy’s Theorem shown later also calls for a quadrilateral inscribed in a circle) is A+1/4 (“Heron’s Formula and Brahmagupta’s Generalization” 1,2).

Background - Pythagoras

A collection of ancient Hindu manuals called the `Sulba-sˆutras gave detailed geometric instructions for the construction of altars. These guides were written between 500 and 200 BC but they draw upon earlier practices and there are records of similar geometrical construction in Indian writings back to 1000 BC (Barrow 73). These writings show that the famous theorem attributed to Pythagoras was clearly known to them. They needed to do intricate geometrical calculations in order to construct their altars to invoke the powers of the gods. Hence, it appears that the theorem attributed to Pythagoras did not originate with him though he proved it in the 6th century BC. Unfortunately, there are no manuscripts or evidence of his proof though many believe his proof was like the so-called “Chinese Proof”. However, there are those who doubt that this is how Pythagoras did it, others who doubt that Pythagoras ever did it, and still others who doubt that Pythagoras even existed (Dunham, Mathematical 91). In all probability Pythagoras was not an independent inventor of the theorem that bears his name, but learned of it during his travels around the Mediterranean region and with the passage of time became strangely associated with it in the way that legends have a tendency to crystallize around famous people. And, like all legends, it contains a grain of truth: it is the ancient link between geometrical theorems and altar sacrifices (Barrow 75).

No matter how and why we call the theorem attributed to Pythagoras, the Pythagorean Theorem is one of the most important theorems in all geometry. It is also important to note that Pythagoras was a great influence and, in fact, to him we owe the very word mathematics and its doubly two-fold branches.

Mathematics

The discrete
The continued

The absolute
The relative

The stable
The moving

Arithmetic
(Newman 85).

Music

Geometry
Astronomy





          Pythagoras, a pupil of Thales, is supposed to have been a native of Samos living from about 584 BC to 495 BC. He lectured on philosophy and mathematics and enthusiastic hearers of all ranks flocked to hear him. Even women, who were banned from public meetings, broke the law to hear him. His followers formed a society and were known as the Order of the Pythagoreans and they exercised a great influence across the Grecian world. The Pythagoreans made great progress in mathematics particularly in the theory of numbers and in the geometry of areas and solids. Much of what was done by the Pythagoreans has been attributed to Pythagoras himself (Newman 82-83). Pythagoras certainly saw the importance of the theorem that bears his name and we can appreciate the application of it in daily endeavors.

The Pythagorean Theorem states that in a right triangle, the sum of the squares of the two legs is equal to the square of the hypotenuse. There are hundreds of published proofs of the Theorem. The Chinese Proof, A Similarity Proof, and President James A. Garfield’s Trapezoidal Proof can be studied in The Mathematical Universe, An Alphabetical Journey Through the Great Proofs, Problems, and Personalities by William Dunham on pages 91-100. The Similarity Proof is the most common proof shown in elementary geometry textbooks today.

Background - Euler

In the study of Euclidean geometry, three beginning terms are undefined: the point, the line and the plane. In the study of arithmetic, a number line is used to study numbers. In algebra, two perpendicular lines (called the cartesian coordinate system set up by René Descartes) are used to denote ordered pairs which is a beginning to study two dimensional curves. When we talk about the shortest distance between two points, we think of a line. Points that lie on a line are called collinear. Lines occur everywhere in our world. When we see things that are collinear, we notice that fact. If we take three line segments such that the sum of the shortest two is greater than the longer, we can make a unique triangle. The perpendicular bisectors of the three sides of the triangle intersect in one point (called the circumcenter). The three altitudes of the triangle intersect in one point (called the orthocenter). The three medians of the triangle intersect in one point (called the centroid, the center of gravity). These three points (the circumcenter, orthocenter and centroid) all lie on one line. Leonhard Euler proved this fact and it is now called Euler’s line.

            In the study of plane figures, we start with the line and then the circle. If we take the midpoint of the segment whose endpoints are the circumcenter and the orthocenter of a triangle, that point is the center of a circle that passes through the following six points: the midpoints of the three sides of the triangle and the point where each of the three altitudes intersect the sides of the triangle (see figure 1). The Nine-Point circle is then drawn (the nine points include the six points on the circle, the center of the circle and the circumcenter and orthocenter of the triangle). This peculiar theorem, overlooked by Euclid, Archimedes, Ptolemy, and everyone else for thousands of years showed that Euler could geometrize with the best of them (Dunham, Mathematical 56).

            Leonhard Euler was born in Basel, Switzerland in 1707. He was one of the most prolific mathematicians of all time. Accomplished in such subjects as number theory, calculus, algebra, and geometry, he also gave birth almost single-handedly to new branches of mathematics such as graph theory, the calculus of variations and combinatorial topology. He was influential in establishing the legitimacy of complex numbers and of elevating to prominence the idea of a function that now unifies much of mathematics. (Dunham, Mathematical 52). Even though elementary geometry had been thoroughly explored by outstanding mathematicians throughout the ages, Euler was able to find new theorems like the Nine-Point Circle mentioned above.

            In a study of constructions in elementary geometry using only a compass and a straightedge, Euler’s line and the Nine-Point Circle can be constructed by all students. For added enrichment, students can look into the justification of these two situations.

Background - Ptolemy

Ptolemy of Alexandria, who lived in the second century AD, wrote the most comprehensive astronomical treatise that has come down to us from antiquity. He was responsible for several innovations in the theory of the movements of the moon and planets (Lloyd 114). In his Mathematical Composition, which is generally referred to by its Arabic name, the Almagest, he uses a lot of points, lines, line segments, circles and arcs to describe models and movement of planets. Although he set out to establish that the earth is at the center of the universe and that the earth does not move, he did specifically record what was believed in his time. He subdivides theoretical studies into three, theology, the study of god (conceived as invisible and unchanging) - physics, the study of the world of change in the sublunary sphere, and mathematics, including, especially, theoretical astronomy. Both theology and physics are matters of conjecture rather than of scientific understanding, theology because of its utter obscurity and physics because of the instability of what it deals with. Mathematics alone yields unshakable knowledge, proceeding as it does by means of indisputable arithmetical and geometrical demonstrations (Lloyd 114). With his belief in the exactness of geometric demonstrations, it follows that he would state a theorem that now bears his name.

Ptolemy’s Theorem states that in an quadrilateral inscribed in a circle, the sum of the products of its opposite sides is equal to the product of its diagonals (see figure 2). This is a theorem than can be proved by advanced students in elementary geometry using similar triangles and their properties. Ptolemy’s Theorem also implies the Pythagorean Theorem (Bogomolnv 1,2). This implication can be discussed with students of trigonometry.

Background - Ceva

Giovanni Ceva was an Italian mathematician who lived from 1647 to 1734. He was Professor of Mathematics at the University of Mantua from 1686 to the end of his life. For most of his life, he worked on geometry. He also proved Ptolemy’s Theorem. He discovered one of the most important results in the synthetic geometry of the triangle between Greek times and the 19th century. This result is called Ceva’s Theorem.   Ceva’s Theorem, described below, picked up on the work of Menelaus of Alexandria (AD 70-120) who contributed greatly to the world of mathematics and science. Ceva also did important work on hydraulics. He held official positions in Mantua and used his knowledge of hydraulics to argue successfully against a project which proposed to divert the river Reno into the river Po (O’Connor & Robertson 1).

Menelaus’ Theorem states that the six segments formed by a transversal cutting the three sides of a triangle (one of the three sides is extended) are such that the product of the three segments having no common endpoint is equal to the product, numerically, of the remaining three (see figure 3).

Ceva’s Theorem states that if three concurrent lines, one from each vertex of a triangle, are drawn, they divide the opposite sides into six segments such that the product of three segments having no common endpoint is equal to the product of the other three segments (see figure 4).

The theorems of Menelaus and Ceva go together since the former gives the condition for points on the sides of a triangle to be collinear and the latter gives the conditions for lines through vertices of a triangle to be concurrent.

It is unfortunate that Ceva’s Theorem is seldom mentioned in elementary geometry courses. It’s a regrettable fact because not only does it unify several other geometric principles but its proof is as simple as that of other less general theorems (Bogomolnv 1). Ceva’s Theorem can proved by advanced students in elementary geometry using the Theorem of Proportionality and Similar Triangles.

Implementation

This unit is designed to be used in parts as the topic comes up in the geometry course.

District Standards:

This unit addresses the following Standards and Benchmarks from the Albuquerque Public Schools 2001 Draft Entitled K-12 Mathematics Content and Performance Standards:

Strand I: Global Mathematical Processes

K-12 Content Standard: The student understands and uses mathematical processes.

K-12 Benchmark: The student uses problem solving, reasoning and proof, communications, connections, and representations as appropriate in all mathematical experiences.

Performance Standards K-12

1.         Develops resourcefulness and perseverance in problem solving in mathematics and other disciplines.

2.         Recognizes when to use previously learned strategies to solve new problems.

5.         Makes and investigates mathematical conjectures and use them successfully in developing and evaluating mathematical arguments and proof.

7.         Develops a logical sequence of arguments leading to a valid conclusion or solution to a problem (statement/reasons, proof, informal proof, and algebraic steps).

12.       Identifies and connects functions with real-world applications.

Strand II: Number Sense and Operations

K-12 Content Standard: The student demonstrates number sense through experiences with meaningful mathematical problems that focus on number meaning, number relationships, place value concepts, relative effects of operations, and multiple representations to communicate sound mathematical thinking.

9-12 Benchmark: The student understands rational, real, and complex numbers and uses a variety of means, including technology, as appropriate, to solve problems in these number systems.

Grade Performance Standards - Early High School

Computation and Estimation

8.         Applies ratios, proportions, and percents in more complex mathematical situations.

Strand III: Geometry, Spatial Sense, and Measurement

K-12 Content Standard: The student demonstrates an understanding of concepts, properties, and relationships of geometry and measurement through experiences with meaningful mathematical problems that focus on identifying, describing, classifying, visualizing, comparing, estimating, and measuring various aspects of shapes and objects.

9-12 Benchmark: The student probes theorems, explores and tests several logical reasoning methods, uses trigonometric relationships and Cartesian coordinates to represent objects in the plane. The student uses formulas for solving measurement problems and uses scaling as appropriate.

Grade Performance Standards - Early High School

Geometric Language and Symbols

1.         Uses and applies commonly used geometric symbols.

2.         States and applies the properties of congruent and similar figures and solids.

Geometric Figures

4.         Draws and constructs representations of 2- and 3-dimensional geometric objects using a variety of tools.

Pythagorean Theorem

5.         Uses the Pythagorean Theorem to solve problems involving right triangles.

Grade Performance Standards - Late High School

Plane and Solid Figures

2.         States and applies the properties of altitudes, medians, angle bisectors, and perpendicular bisectors of the sides of triangles, angles, and parallel lines.

Geometric Figures

9.         Uses compass and straightedge to perform basic constructions using appropriate units and scales of measurement.

Proof

17.       Makes and investigates geometric conjectures and uses them successfully in developing and evaluating mathematical arguments and proofs.

Implementation - Heron

Materials:        Pencil
                                    Ruler
                                    Straightedge
                                    Compass
                                    Paper with the lengths of three line segments marked
                                    Scissors
                                    Calculator

Procedure:

Tell students that we are going to compute the area of a triangle given the lengths of its sides. Pass out materials and have students construct a triangle given that the lengths of the three sides are the lengths of the three line segments given. (Students will have already been instructed on how to do geometric construction) Have them cut out the triangle and then compare their triangle with the triangles cut out by other students. Ask what is true about all the triangles and why. They will say that they are congruent because of the Postulate Side-Side-Side. This will be a good quick review of Postulates and Congruent Triangles. Ask students to find the approximate area of the triangle that they just constructed using the ruler as a measure. They will, most likely, find an altitude and use the formula Area = 1/2 base times height. Have students use each side as a base and come up with three answers (which, of course, would be equal to each other). Ask: “If you know the 3 sides of a large triangular field, how practical would it be to have to measure an altitude in order to find the area of the field.” Can we find the area without finding an altitude? Since we will have already covered the sine, cosine, and tangent ratios in this geometry class, some students may try to come up with the altitude using the sine ratio. They, however, would need to use the Law of Cosines (which they haven’t covered) to find an appropriate angle. And, in order to use the sine ratio, one usually needs a trigonometry table or a scientific calculator. Since we have only been given the measures of the three sides, ask students what information should the formula only ask for in order for it to be most easily used. Ask questions until someone comes up with the idea that it would be nice to only have to use the measure of the three sides directly in the formula. Then ask what mathematical operations they first learned as they were growing up (add, subtract, multiply, divide, and square root). Heron of Alexandria thought just that and worked geometrically to come up with the formula that bears his name. Students too can work to come up with theorems and formulas to make their work easier. Since most students want to know how they are going to use what they learn, it is important to see the thinking process that goes to make things easier.

            Let the three sides be a, b, and c. Ask what the length a + b + c is called (perimeter). Ask what “half” a circle is called (semi-circle). What could one-half of (a + b + c) be called (semi-perimeter)? What is the first letter of “semi-perimeter” (s)? State the formula showing that only a, b, c, and s are used. Then have students compute the area of the triangle that they constructed using Heron’s Formula where s=1/2(a+b+c) and compare it with the approximation they got with the ruler and the formula A= 1/2bh. State that it is possible to find the square root to find the area without the use of a calculator. State that their parents and grandparents were taught how to find the square root using the square root algorithm when they were in the eighth grade.

Assessment:

Have students measure certain triangular areas around campus and find the area. Have them compare answers with each other and to the actual area. Have them find polygonal areas of more than 3 sides by dividing them into triangles and using Heron’s Formula.

Follow-up:

Have students find triangular areas of their own and find the area. Have them describe why it might be easier to use Heron’s formula rather than finding an altitude. They could find the area in both ways and show the difference in the work done. Have them research Heron and Heron’s formula and do a report on the development and/or proof of the theorem. This can be a topic for their oral report. Have students find out how to use the square root algorithm and its proof. This can also be a topic for their oral report.

Implementation - Pythagoras

Materials:        Pencil
                                    Ruler
                                    Straightedge
                                    Compass
                                    Paper with the lengths of six line segments measuring 6 inches, 8 inches, 5                                       centimeters, 12 centimeters, 5 inches and 7 inches
                                    Scissors

Procedure:

Tell students that we are going to start by constructing a right triangle given the lengths of two legs. Using the given 6 inch and 8 inch segments, construct a right triangle with legs measuring 6 inches and 8 inches. Construct the hypotenuse. Measure the length of the hypotenuse with the ruler. They should get 10 inches. Write down the 3 sides of the triangle from shortest to longest (6, 8, 10). Have students try to find a relationship between the three numbers. Most of the students will have learned the relationship between the three sides of a right triangle in middle school, but this is an exercise to show them that it actually works. Have the students do the same with 5 centimeters and 12 centimeters. They should get 5, 12, 13. Have students write the relationship of these 3 numbers and compare it with 6, 8, 10. Mention to the students that since all three numbers are integers, they are called Pythagorean Triples. Have students now do the same with 5 inches and 7 inches. Have them write down the measure of the third side. In this case the third side is not an integer. Show how the Pythagorean Theorem works using the first two examples. Then use the it to find the hypotenuse of the third and compare the answer with the measured approximation of the hypotenuse.

Using similar triangles and the geometric mean, lead students into the proof of the Pythagorean Theorem. Tell students about Pythagoras and mention that there are hundreds of proofs of this theorem.

Assessment:

Have students find the 3rd side of a right triangle given 2 of its sides. There are plenty of these problems in their textbook and more practical problems can be given (like if Santa Fe were due north of Albuquerque and Gallup were due west of Albuquerque, find the distance from Santa Fe to Gallup given the distances to Santa Fe and Gallup from Albuquerque.)

Follow-up:

Have students make up problems using the Pythagorean Theorem. The best of these could be used on the next test or on the final exam. Have students research some of the proofs of the theorem. This can be one of the topics for their oral presentation.

Implementation -- Euler

Materials:        Pencil
                                    Compass
                                    Straightedge
                                    Two papers each with a large scalene triangle drawn on it

Procedure:

Give students a sheet of paper wherein is drawn a large scalene triangle. Have them construct the perpendicular bisectors of the three sides and label the point of intersection “A” (the 3 lines should be concurrent in a point called the circumcenter). Have them construct the altitudes of the three sides of the triangle and label the point of intersection “B” (the 3 lines should be concurrent in a point called the orthocenter). Have them construct the three medians of the triangle and label the point of intersection “C” (the 3 lines should be concurrent in a point called the centroid). Discuss the words circumcenter, orthocenter, and centroid to show how the points got those names. Ask what appears to be true about the points A, B, and C? Have student compare their graphs and discuss what appears to be true about the 3 points. Mention that this line is called Euler’s Line and talk about Euler’s prolific contributions to mathematics and the fact that he came up with this line and the following Nine-Point Circle long after most of Euclidean Geometry had been developed.

Assessment:

Mention that constructions need to be done very carefully and if they got Euler’s line perfectly, they have done a careful job. To see how carefully they are doing constructions, give the students another paper with the same large scalene triangle drawn on it. Have them construct the midpoint of the segment whose endpoints are the circumcenter and the orthocenter of the triangle. Then, using this midpoint as the center of a circle, draw a circle that passes through the midpoint of one side of the original triangle. This circle should, then, also pass through the midpoints of the other two sides and the points where each of the three altitudes intersect the sides of the triangle (see figure 1).

Follow-up:

Students can research the justification of Euler’s line or Nine-Point Circle and report on it to the class.

Implementation - Ptolemy

Materials:        Pencil
                                    Compass
                                    Straightedge
                                    Ruler
                                    Calculator

Procedure:

Mention that the earth is the center of the universe and that the earth does not move. State that in Ptolemy’s time (second century AD) that this belief was wide spread and he recorded this geometrically in wonderful terms and diagrams. Even though his astronomical beliefs needed some revision, Ptolemy did produce many true geometrical demonstrations. Have students construct a quadrilateral inscribed in a circle labeling the vertices A, B, C, and D. Draw the diagonals AC and DB (see figure 2). Using a ruler, measure the lengths of the sides of the quadrilateral (find AB, BC, CD, and DA) and of the diagonals (find AC and DB). Find the value of AB*DC , AD*BC, and AC*BD. Have them come out with a relationship between these three values. Do this for another quadrilateral inscribed in a circle. Have students come up with a rule about these three values. State that this is Ptolemy’s Theorem.

Assessment and Followup:

Have students prove this theorem using similar triangles and their properties. This can be used for an oral presentation or as an extra credit assignment.

Implementation - Ceva

Materials:        Diagram of Menelaus’ Theorem (figure 3)
                                    Pencil
                                    Paper
                                    Ruler
                                    Calculator

State that an Italian mathematician named Ceva who lived from the late 17th century to the early 18th century studied ancient mathematicians and proved Ptolemy’s Theorem. Give out a diagram of Menelaus’ Theorem (figure 3) and state that he was a 2nd century AD mathematician and that Ceva continued the work that Menelaus started. Describe Menelaus’ Theorem and discuss its proof with the students (Davis 1,2). Give students a sheet of paper with a large scalene triangle drawn on it. Label the vertices A , B , and C. Choose a point interior to the triangle and label the point P. Draw a line segment from A through P and extend it until it intersects CB. Name the point of intersection Y. Draw a line segment from B through P and extend it until it intersects AC. Name the point of intersection Z. Draw a line segment from C through P and extend it until it intersects AB. Name the point of intersection X (see figure 4). Have students find AX, BY, CZ, XB, YC, and ZA. Then find AX?BY?CZ and XB?YC?ZA. Compare the result and have students use the result to come up with a product of three ratios that equal one.   Use this result to state Ceva’s Theorem in words. Have students look at the similarities between Menelaus’ Theorem and Ceva’s Theorem. Ask students what Ceva may have seen from Menelaus’ Theorem and how he may have come up with his own theorem about sixteen centuries later.

Assessment and follow-up:

Have students use the Theorem of Proportionality and Similar Triangles to prove Ceva’s Theorem. This can be used for an oral presentation or as an extra credit assignment.

Documentation

Albuquerque Public Schools, Mathematics Course Offerings, Geometry Honors, 2000.

Barrow, John D. Pi in the Sky. Oxford: Oxford University Press, 1992

Bogomolnv, Alexander. “Sine, Cosine, and Ptolemy’s Theorem.” 1996-2002
<http://www.cut-the-knot.com/proofs/sine_cosine.shtm l>.

Davis, Hamilton. “Menelaus’ and Ceva’s Theorem and their many applications.” 2001
            <http://hamiltonious.virtualave.net/essays/othe/finalpaper4.htm>

Dunham, William. Journey Through Genius The Great Theorems of Mathematics. New York,              New York: John Wiley & Sons, Inc., 1990.

---. The Mathematical Universe An Alphabetical Journey Through the Great Proofs, Problems, and Personalities. New York, New York: John Wiley & Sons, 1994.

“Heron’s Formula and Brahmagupta’s Generalization.”
            <http://www.mathpages.com/home/kmath196.htm>

Lloyd, G. E. R. Greek Science After Aristotle. New York, New York: W.W. Norton                      & Company, Inc., 1973.

Newman, James R. The World of Mathematics. New York, New York: Simon and                         Schuster, 1956.

O’Conner, J.J. and Robertson, E.F. “Giovanni Ceva.” JOC/EFR. December 1996. School of              Mathematics and Statistics, University of St. Andrews, Scotland.
           http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Ceva_Giovanni.html

Figure 1 - Euler’s Nine-Point Circle

Given ABC.

M1, M2 and M3 are the midpoints of AB, BC and AC, respectively. H1, H2 and H3 are the points where the altitudes from points C, A and B, respectively, intersect the sides of the triangle.X is the circumcenter and Y is the orthocenter of ABC. O is the midpoint of XY and is also the center of the circle that contains M1, M2, M3, H1, H2 and H3.

Figure 2 - Diagram for Ptolemy’s Theorem

Ptolemy’s Theorem - If a quadrilateral is inscribed in a circle, the sum of the products of its opposite sides is equal to the product of its diagonals.

AB * DC + AD * BC   =    AC * DB


            Figure 3 - Diagram for Menelaus’ Theorem

Menelaus’ Theorem: The six segments formed by a transversal cutting the three sides of a triangle or a side of a triangle extended (BC is the side extended) are such that the product of the three segments having no common endpoint is equal to the product, numerically, of the remaining three.
NC * MA * BL = AN * BM * LC     or     **=1



                                                                                                                                                                   Figure 4 - Diagram for Ceva’s Theorem

Ceva’s Theorem - If three concurrent lines, one from each vertex of a triangle, are drawn, they divide the opposite sides into six segments such that the product of three segments having no common endpoint is equal to the product of the other three segments.

If P is any point inside ABC, then AX*BY*CZ = XB*YC*ZA

or **