Mathematics
Circle Theorems
Circle theorems are a set of rules and principles that apply to circles and their properties. They are used to solve problems involving angles, chords, tangents, and other geometric elements within circles. These theorems are fundamental in geometry and are essential for understanding and solving problems related to circles.
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7 Key excerpts on "Circle Theorems"
eBook - PDF- Doug French(Author)
- 2004(Publication Date)
- Continuum(Publisher)
A ring of four circles are constructed so that the outer set of four intersection points lie on a common circle. The remarkable fact is that the inner Figure 9.21 Miquel's six-circle theorem The Circle 117 set of four intersection points also lie on a circle. It is fascinating to create the configuration with dynamic geometry software and observe the effect of varying the circles. The proof is surprisingly simple and elegant. Sets of four intersection points form quadri-laterals, which are shown in Figure 9.22 without the circumscribing circles. If the four outer quadrilaterals are cyclic then the four pairs of angles denoted by a, b, c and d are equal because of the exterior angle property. If the outer quadrilateral is cyclic, then a + b + c + d=8Q° from which it immediately follows that the inner quadrilateral is cyclic also. Figure 9.22 Six cyclic quadrilaterals As a final bonus, if the centres of the initial ring of four circles lie on the outer circle, then the inner quadrilateral is a rectangle, a result that readers are invited to explore for themselves. CONCLUSION The Circle Theorems provide an excellent example of a family of linked results which provide a valuable source of examples of both traditional angle-finding problems and surprising results which encourage a search for proofs. They do not provide a lot of immediately obvious real-world applications, but they are rich in intrinsic interest and have a way of arising in unlikely ways like proving trigonometrical identities or finding the circumference of a fifty-pence coin. Learning to work with the Circle Theorems brings together many aspects of learning to think geometrically. Success in solving problems and appreciating and generating proofs requires both a knowledge of facts and a range of strategies for producing appropriate chains of reasoning.
eBook - PDF- H. S. M. Coxeter, S. L. Greitzer(Authors)
- 1967(Publication Date)
- American Mathematical Society(Publisher)
C H A P T E R 2 Some Properties of Circles Although the Greeks worked fruitfully, not only in geometry but also in the most varied fields of mathematics, neverthe- less we today have gone beyond them everywhere and cer- tainly also in geometry. F. KIcin The circle has been held in highest esteem through the ages. Its perfect form has affected philosophers and astronomers alike. Until Kepler derived his laws, the thought that planets might move in anything but circular paths was unthinkable. Nowadays, the words “square”, “line”, and the like sometimes have derogatory connotations, but the circle- never. Cleared of superstitious nonsense and pseudo-science, it still stands out, as estimable as ever. Limitations of space make it impossible for us to present more than a few of the most interesting properties developed since Euclid of the circle and its relation to triangles and other polygons. 2.1 The power of a point with respect to a circle We begin our investigations by recalling two of Euclid’s theorems: 111.35, about the product of the parts into which two chords of a circle divide each other (that is, in the notation of Figure 2.1A, PA X PA’ = PB X PB’ ), and 111.36, comparing a. secant and a tangent drawn from the same point P outside the circle (in Figure 2.1B, PA X PA’ = PTS ). If we agree to regard a tangent as the limiting form of a secant, we can combine these results as follows: 27 28 PROPERTIES OF CIRCLES THEOREM 2.11. If two lines through a point P meet a circle at points A, A' (possibly coincident) and B, B' (possibly coincident), respec- tively, then PA X PA' = PB X PB'. - Figure 2.1A For a proof we merely have to observe that the similar triangles PAB' and PBA' (with a common angle at P ) yield PA PB PB' PA -= -In Figure 2.1B, we can equally well use the similar triangles PAT and PA PT PTA' to obtain ----PT PA and then say PA X PA' = P P = PB X PB'. - Figure 2.1B Let R denote the radius of the circle, and d the distance from P to the center.
eBook - PDFCollege Geometry
Using the Geometer's Sketchpad
- Barbara E. Reynolds, William E. Fenton(Authors)
- 2012(Publication Date)
- Wiley(Publisher)
At the very least, the experimentation you have done in constructing the diagram will have challenged you to think about the ideas you will need as you develop a proof. In these first several chapters we have been working with Euclid’s axioms. In some later chapters, we will change one of these axioms and observe the effects of that change. This will give us an opportunity to see how much our world view is shaped by our assumptions. (In other words, what we see depends on our axioms.) The geometric content of this chapter has been focused on properties of cir- cles. You have been reminded of a lot of the terminology used in talking about circles. You have probably also encountered some new and challenging ideas in the geometry of circles. A triangle has many circles associated with it, and this provides a bridge back to the work of the previous chapter. Each triangle has one incircle and three excircles, which are associated with the bisectors of the inter- nal and external angles of the triangle. The circumcircle passes through the three vertices of the triangle, and its center can be found by constructing the perpendic- ular bisectors of the sides. The nine-point circle of a triangle passes through many interesting points. We will continue our investigations of the nine-point circle in the next chapter, where we will be able to use methods of analytic geometry to develop some of the proofs. This chapter also explored cyclic quadrilaterals and some of the special properties they possess. We can construct various families of circles—circles that share particular characteristics. For example, there is a family of circles that all have the same center point O, another family of circles that all pass through a point A, and a family of circles with the same radius. We have investigated two interesting families of circles in this chapter: circles that share a common chord, and circles that are orthogonal to a given fixed circle.
eBook - PDF- Barbara E. Reynolds, William E. Fenton(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
We will continue our investigations of this special circle in Chapter 5, Analytic Geometry, where you will have an opportunity to use methods of analytic geometry to develop some of the proofs [Thomas 2002, 69; Wells 1991, 76 & 159]. 4.3 E X E R C I S E S Give clear and complete answers to the following problems and questions. Write your explanations clearly using complete sentences. Include diagrams whenever appropriate. 1. The following statements have all appeared in the activities or exercises of this or the previous chapters. You have already constructed GeoGebra diagrams illustrating these statements, and may already have written informal proofs of some of these statements. Write out a careful step-by-step proof for each statement. a. Let O be the center of a circle, and let P, Q, and R be points on the circle. Prove that the measure of the central angle ∠POR is twice the measure of the inscribed angle ∠PQR. b. Let O be the center of a circle, and let PR be a diameter of this circle. If Q is a point on the circle, prove that angle ∠PQR is a right angle. c. A median divides its triangle into two equal areas. d. The three medians of a triangle are concurrent at a point called the centroid, often denoted as G. EXERCISES 103 e. The three altitudes of a triangle are concurrent at a point called the orthocenter, often denoted as H. f. The perpendicular bisectors of the three sides of a triangle are concurrent at a point called the circumcenter, often denoted as O. g. The circumcenter, O, of a triangle is the center of a circle that passes through the three vertices of the triangle. h. The opposite interior angles of a convex cyclic quadrilateral are supplementary. 2. Create truth tables to verify that the contrapositive, ¬Q → ¬P, and the method of proof by contradiction, (P ∧ ¬Q) → false, are each equivalent to the implication P → Q. 3. Let AB and CD be chords of a circle that intersect at point P. Prove that AP ⋅ PB = CP ⋅ PD. 4. Prove or disprove that a.
eBook - ePubIbn al-Haytham's Geometrical Methods and the Philosophy of Mathematics
A History of Arabic Sciences and Mathematics Volume 5
- Roshdi Rashed(Author)
- 2017(Publication Date)
- Routledge(Publisher)
CHAPTER I THE PROPERTIES OF THE CIRCLE INTRODUCTION The long list of mathematical works written by Ibn al-Haytham includes several that are still missing. Among these are three that, in the mathematics of infinitesimals, speak for themselves: The Greatest Line that can be Drawn in a Segment of a Circle, a Treatise on Centres of Gravity and a Treatise on the Qarasṭūn. These treatises are all concerned with the geometry of measure. Their absence not only deprives historians of mathematics of facts that would have helped them to appreciate more clearly the range of Ibn al-Haytham’s oeuvre, but also, more seriously, it makes it absolutely impossible for them to understand the structures of this oeuvre and the network of meanings that they carry. If the first of the treatises cited above had been at our disposal, we should have a better understanding of the distance the author of a treatise on problems of figures with equal perimeters, on figures with equal areas and on the solid angle, travelled along the road of what was later to be called the calculus of variations. This state of affairs is not peculiar to the geometry of measure; it is found also in the other type of geometry developed by Ibn al-Haytham and his predecessors: the geometry of position and forms. Among the books that until very recently were still missing we have one with the title On the Properties of Circles. A book with such a title is of course intriguing and surprising. 1 We ask ourselves what Ibn al-Haytham might deal with in a book whose title appears so strikingly modern. His predecessors, his contemporaries and Ibn al-Haytham himself wrote books and papers on one or another aspect of a geometrical figure, for example triangles, but rarely on all its properties taken together as a whole. Furthermore, Ibn al-Haytham had written more than once on the circle, on finding its perimeter and finding its area
eBook - PDF- Patrick D Barry(Author)
- 2001(Publication Date)
- Woodhead Publishing(Publisher)
7 Circles; their basic properties Hitherto our sets have involved lines and half-planes, and specific subsets of these. Now we introduce circles and study their relationships to lines. We do not do this just to admire the circles, and to behold their striking properties of symmetry. They are the means by which we control angles, and simplify our work on them. 7.1 INTERSECTION OF A LINE AND A CIRCLE 7.1.1 Terminology concerning a circle Definition . If O is any point of the plane Π and k is any positive real number, we call the set C ( O ; k ) of all points X in Π which are at a distance k from O , i.e. C ( O ; k ) = { X ∈ Π : | O , X | = k } , the circle with centre O and length of radius k . If X ∈ C ( O ; k ) the segment [ O, X ] is called a radius of the circle. Any point U such that | O , U | < k is said to be an interior point for this circle. Any point V such that | O , V | > k is said to be an exterior point for this circle. For every circle C ( O ; k ) and line l , one of the following holds:-(i) l ∩ C ( O ; k ) = { P } for some point P , in which case every point of l { P } is exterior to the circle. (ii) l ∩C ( O ; k ) = { P, Q } for some points P and Q , with P = Q , in which case every point of [ P, Q ] { P, Q } is interior to the circle, and every point of PQ [ P, Q ] is exterior to the circle. (iii) l ∩ C ( O ; k ) = ∅ , in which case every point of l is exterior to the circle. Proof . Let M = π l ( O ) , and let m be the line which contains M and is perpendicular to l , so that O ∈ m . 105
eBook - PDF- Paul Groth, Todd W. Bressi(Authors)
- 2006(Publication Date)
- Yale University Press(Publisher)
SECTION VII EQUATIONS DEFINING SECTIONS OF A CIRCLE ^335. Among the splendid developments contributed by modern mathematicians, the theory of circular functions without doubt holds a most important place. We shall have occasion in a variety of contexts to refer to this remarkable type of quantity, and there is no part of general mathematics that does not depend on it in some fashion. Since the most brilliant modern mathematicians by their industry and shrewdness have formulated for it an extensive discipline, we can hardly expect that any part of the theory, particularly the elements, can be significantly expanded. I will speak of the theory of trigonometric functions as related to arcs that are commensurable with the circumference, or of the theory of regular polygons. Only a small part of this theory has been developed so far, as the present section will make clear. The reader might be surprised to find a discussion of this subject in the present work which deals with a discipline apparently so unrelated; but the treatment itself will make abundantly clear that there is an intimate connection between this subject and higher Arithmetic. The principles of the theory which we are going to explain actually extend much farther than we will indicate. For they can be applied not only to circular functions but just as well to other transcendental functions, e.g. to those which depend on the integral J[ !/>/(! — x 4 )]dx and also to various types of con-gruences. Since, however, we are preparing a substantial work on transcendental functions and since we will treat congruences at length as we continue our discussion of arithmetic, we have decided to consider only circular functions here. And although we could discuss them in all their generality, we reduce them to the simplest case in the following article, both for the sake of brevity and in order that the new principles of this theory may be more easily understood. 407
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