A Theorem on Circle Configurations 1. Introduction
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10.4 - Circles, Arcs, Inscribed Angles, Power of a Point We work in Euclidean Geometry for the remainder of these notes. Definition: A minor arc is the intersection of a circle with a central angle and its interior. A semicircle is the intersection of a circle with a closed half-plane whose boundary line passes through its center. A major arc is the intersection of a circle with a central angle and its exterior (that is, the complement of a minor arc, plus its endpoints). Notation: If the endpoints of an arc are A and B, and C is any other point of the arc (which must be used in order to uniquely identify the arc) then we denote such an arc by qACB . Measures of Arcs: We define the measure of an arc as follows: Minor Arc Semicircle Major Arc mqACB = µ(pAOB) mqACB = 180 m=qACB 360 - µ(pAOB) A A A C O O C O C B B B q q Theorem (Additivity of Arc Measure): Suppose arcs A1 = APB and A2 =BQC are any two arcs of a circle with center O having just one point B in common, and such that their union A1 c q A2 = ABC is also an arc. Then m(A1 c A2) = mA1 + mA2. The proof is straightforward and tedious, falling into cases depending on whether qABC is a minor arc or a semicircle, or a major arc. It is mainly about algebra and offers no new insights. We’ll leave it to others who have boundless energy and time, and who can wring no more entertainment out of “24.” Lemma: If pABC is an inscribed angle of a circle O and the center of the circle lies on one of its 1 sides, then μ()∠=ABC mACp . -
Power of a Point Yufei Zhao
Trinity Training 2011 Power of a Point Yufei Zhao Power of a Point Yufei Zhao Trinity College, Cambridge [email protected] April 2011 Power of a point is a frequently used tool in Olympiad geometry. Theorem 1 (Power of a point). Let Γ be a circle, and P a point. Let a line through P meet Γ at points A and B, and let another line through P meet Γ at points C and D. Then PA · PB = PC · P D: A B C A P B D P D C Figure 1: Power of a point. Proof. There are two configurations to consider, depending on whether P lies inside the circle or outside the circle. In the case when P lies inside the circle, as the left diagram in Figure 1, we have \P AD = \PCB and \AP D = \CPB, so that triangles P AD and PCB are similar PA PC (note the order of the vertices). Hence PD = PB . Rearranging we get PA · PB = PC · PD. When P lies outside the circle, as in the right diagram in Figure 1, we have \P AD = \PCB and \AP D = \CPB, so again triangles P AD and PCB are similar. We get the same result in this case. As a special case, when P lies outside the circle, and PC is a tangent, as in Figure 2, we have PA · PB = PC2: B A P C Figure 2: PA · PB = PC2 The theorem has a useful converse for proving that four points are concyclic. 1 Trinity Training 2011 Power of a Point Yufei Zhao Theorem 2 (Converse to power of a point). -
Group 3: Clint Chan Karen Duncan Jake Jarvis Matt Johnson “The
Group 3: Clint Chan Karen Duncan Jake Jarvis Matt Johnson “The power of a point P with respect to a circle is the product of the two distances PA times PB from the point to the circle measured along a random secant” (B&B 261). When the point P is on the circle, its power is zero; when it is inside the circle, its power is negative; and when it is outside the circle, its power is positive. The power of a point when P is outside the circle is also equal to (PT)^2, where T is the point where the line through P is tangent to the circle. Also of interest, if P is outside the circle and e is a circle which is orthogonal to c and centered at P, then pc(A) = t^2, where t is the radius of e (from "The Power of a Point and Radical Axis" class notes). The radical axis of two circles is, by definition, the locus of points A for which the power of A with respect to c[1] and c[2] are equal. Using some facts about the power of a point, we can see that the radical axis is a line. Fact 1: If A is a point outside the circle and S is a secant intersecting c at M and N, then the power of A with respect to c is equal to _AM__AN_. Fact 2: If A is a point outside the circle and AT s a line tangent to c at T, then the power of A with respect to c is equal to _AT_^2. -
Apollonian Circle Packings: Dynamics and Number Theory
APOLLONIAN CIRCLE PACKINGS: DYNAMICS AND NUMBER THEORY HEE OH Abstract. We give an overview of various counting problems for Apol- lonian circle packings, which turn out to be related to problems in dy- namics and number theory for thin groups. This survey article is an expanded version of my lecture notes prepared for the 13th Takagi lec- tures given at RIMS, Kyoto in the fall of 2013. Contents 1. Counting problems for Apollonian circle packings 1 2. Hidden symmetries and Orbital counting problem 7 3. Counting, Mixing, and the Bowen-Margulis-Sullivan measure 9 4. Integral Apollonian circle packings 15 5. Expanders and Sieve 19 References 25 1. Counting problems for Apollonian circle packings An Apollonian circle packing is one of the most of beautiful circle packings whose construction can be described in a very simple manner based on an old theorem of Apollonius of Perga: Theorem 1.1 (Apollonius of Perga, 262-190 BC). Given 3 mutually tangent circles in the plane, there exist exactly two circles tangent to all three. Figure 1. Pictorial proof of the Apollonius theorem 1 2 HEE OH Figure 2. Possible configurations of four mutually tangent circles Proof. We give a modern proof, using the linear fractional transformations ^ of PSL2(C) on the extended complex plane C = C [ f1g, known as M¨obius transformations: a b az + b (z) = ; c d cz + d where a; b; c; d 2 C with ad − bc = 1 and z 2 C [ f1g. As is well known, a M¨obiustransformation maps circles in C^ to circles in C^, preserving angles between them. -
Apollonius of Pergaconics. Books One - Seven
APOLLONIUS OF PERGACONICS. BOOKS ONE - SEVEN INTRODUCTION A. Apollonius at Perga Apollonius was born at Perga (Περγα) on the Southern coast of Asia Mi- nor, near the modern Turkish city of Bursa. Little is known about his life before he arrived in Alexandria, where he studied. Certain information about Apollonius’ life in Asia Minor can be obtained from his preface to Book 2 of Conics. The name “Apollonius”(Apollonius) means “devoted to Apollo”, similarly to “Artemius” or “Demetrius” meaning “devoted to Artemis or Demeter”. In the mentioned preface Apollonius writes to Eudemus of Pergamum that he sends him one of the books of Conics via his son also named Apollonius. The coincidence shows that this name was traditional in the family, and in all prob- ability Apollonius’ ancestors were priests of Apollo. Asia Minor during many centuries was for Indo-European tribes a bridge to Europe from their pre-fatherland south of the Caspian Sea. The Indo-European nation living in Asia Minor in 2nd and the beginning of the 1st millennia B.C. was usually called Hittites. Hittites are mentioned in the Bible and in Egyptian papyri. A military leader serving under the Biblical king David was the Hittite Uriah. His wife Bath- sheba, after his death, became the wife of king David and the mother of king Solomon. Hittites had a cuneiform writing analogous to the Babylonian one and hi- eroglyphs analogous to Egyptian ones. The Czech historian Bedrich Hrozny (1879-1952) who has deciphered Hittite cuneiform writing had established that the Hittite language belonged to the Western group of Indo-European languages [Hro]. -
MATH 9 HOMEWORK ASSIGNMENT GIVEN on FEBRUARY 3, 2019. CIRCLE INVERSION Circle Inversion Circle Inversion (Or Simply Inversion) I
MATH 9 HOMEWORK ASSIGNMENT GIVEN ON FEBRUARY 3, 2019. CIRCLE INVERSION Circle inversion Circle inversion (or simply inversion) is a geometric transformation of the plane. One can think of the inversion as of reflection with respect to a circle which is analogous to a reflection with respect to a straight line. See wikipedia article on Inversive geometry. Definition of inversion Definition. Given a circle S with the center at point O and having a radius R consider the transformation of the plane taking each point P to a point P 0 such that (i) the image P 0 belongs to a ray OP (ii) the distance jOP 0j satisfies jOP j · jOP 0j = R2. Remark. Strictly speaking the inversion is the transformation not of the whole plane but a plane without point O. The point O does not have an image. It is convenient to think of an “infinitely remote" point O0 added to a euclidian plane so that O and O0 are mapped to each other by the inversion. Basic properties of inversion We formulated and proved some basic properties of the inversion. The most important properties are that the inversion • maps circles and straight lines onto circles and straight lines • preserves angles These properties are stated as homework problems below. Try to prove them indepen- dently. In the following we denote the center of inversion as O, the circle of inversion as S and we use prime to denote the images of various objects under inversion. Classwork Problems The properties presented in problems 1-3 are almost obvious. -
Power of a Point and Radical Axis
Power of a Point and Radical Axis Tovi Wen NYC Math Team §1 Power of a Point This handout will cover the topic power of a point, and one of its more powerful uses in radical axes. Definition (Power of a Point) Given a circle ! with center O and radius r, and a point P , the power of P with respect to !, which we will denote as (P; !) is OP 2 − r2. Note that • If P is outside ! then (P; !) is positive. • If P is on ! then (P; !) = 0. • If P is inside ! then (P; !) is negative. This definition does not feel particularly motivated, and therefore when taught at a more elementary level, it is often skipped and replaced by the following nice property. Theorem Let ! be a circle and let P be a point not on !. If a line passing through P meets ! at distinct points A and B then ( PA · PB if P lies outside !; (P; !) = −PA · PB if P lies inside ! Proof. It is not immediately obvious why the quantity PA · PB should be fixed for any line passing through P . Draw another chord of ! passing through P as shown. B A ω P O C M D Recall that opposite angles in a cyclic quadrilateral are supplementary. This gives \P AC = \P DB so as \AP C ≡ \BP D is shared, we have 4P AC ∼ 4P DB. In particular, PA PD = =) PA · PB = PC · P D: PC PB 1 Power of a Point and Radical Axis Tovi Wen We now show this quantity is equal to OP 2 − r2. -
Power of a Point and Ceva's Theorem
Mathematical Problem Solving Power of a Point A rather simple definition of the power of a point with respect to a circle is: Let C be a circle of radius r. The power of a point P with respect to C is given by d2 − r2, where d is the distance of P to the center of the circle. Just to fix ideas, for example, if C is the circle of radius r centered at the origin, and P has coordinates (x, y), then the power of P with respect to this circle is x2 + y2 − r2. If P is a point, C a circle, I’ll write Π(P, C) to denote the power of P with respect to C. Obviously, Π(P, C) > 0 if and only if P is outside the circle, < 0 if and only if P is inside the circle, and 0 if and only if P is on the circle. The significance of this concept is due to the following result. Theorem 1 Let P,X,X0 be collinear points, let C be a circle. If X,X0 lie on C, and either X 6= X0, or if X = X0 6= P and the line containing X and P is tangent to C, then 0 Π(P, C)= PX · PX , where PX,PX0 are “directed lengths;” to be specific: PX · PX0 < 0 if P is strictly between X,X0, > 0 if either X is strictly between P and X0 or X0 is strictly between P and X, 0 if P coincides with X or X0. -
On a Construction of Hagge
Forum Geometricorum b Volume 7 (2007) 231–247. b b FORUM GEOM ISSN 1534-1178 On a Construction of Hagge Christopher J. Bradley and Geoff C. Smith Abstract. In 1907 Hagge constructed a circle associated with each cevian point P of triangle ABC. If P is on the circumcircle this circle degenerates to a straight line through the orthocenter which is parallel to the Wallace-Simson line of P . We give a new proof of Hagge’s result by a method based on reflections. We introduce an axis associated with the construction, and (via an areal anal- ysis) a conic which generalizes the nine-point circle. The precise locus of the orthocenter in a Brocard porism is identified by using Hagge’s theorem as a tool. Other natural loci associated with Hagge’s construction are discussed. 1. Introduction One hundred years ago, Karl Hagge wrote an article in Zeitschrift fur¨ Mathema- tischen und Naturwissenschaftliche Unterricht entitled (in loose translation) “The Fuhrmann and Brocard circles as special cases of a general circle construction” [5]. In this paper he managed to find an elegant extension of the Wallace-Simson theorem when the generating point is not on the circumcircle. Instead of creating a line, one makes a circle through seven important points. In 2 we give a new proof of the correctness of Hagge’s construction, extend and appl§ y the idea in various ways. As a tribute to Hagge’s beautiful insight, we present this work as a cente- nary celebration. Note that the name Hagge is also associated with other circles [6], but here we refer only to the construction just described. -
Prove It Classroom
How to Prove it ClassRoom SHAILESH SHIRALI Euler’s formula for the area of a pedal triangle Given a triangle ABC and a point P in the plane of ABC In this episode of “How (note that P does not have to lie within the triangle), the To Prove It”, we prove pedal triangle of P with respect to ABC is the triangle △ a beautiful and striking whose vertices are the feet of the perpendiculars drawn from formula first found by P to the sides of ABC. See Figure 1. The pedal triangle relates Leonhard Euler; it gives the area of the pedal triangle in a natural way to the parent triangle, and we may wonder of a point with reference whether there is a convenient formula giving the area of the to another triangle. pedal triangle in terms of the parameters of the parent triangle. The great 18th-century mathematician Euler found just such a formula (given in Box 1). It is a compact and pleasing result, and it expresses the area of the pedal triangle in terms of the radius R of the circumcircle of ABC and the △ distance between P and the centre O of the circumcircle. A O: circumcentre of ABC E △ P: arbitrary point Area ( DEF) 1 OP2 F △ = 1 P Area ( ABC) 4 − R2 O △ ( ) B D C Figure 1. Euler's formula for the area of the pedal triangle of an arbitrary point Keywords: Circle theorem, pedal triangle, power of a point, Euler, sine rule, extended sine rule, Wallace-Simson theorem Azim Premji University At Right Angles, July 2018 95 1 A E F P O B D C Figure 2. -
Efficiently Constructing Tangent Circles
Mathematics Magazine ISSN: 0025-570X (Print) 1930-0980 (Online) Journal homepage: https://www.tandfonline.com/loi/umma20 Efficiently Constructing Tangent Circles Arthur Baragar & Alex Kontorovich To cite this article: Arthur Baragar & Alex Kontorovich (2020) Efficiently Constructing Tangent Circles, Mathematics Magazine, 93:1, 27-32, DOI: 10.1080/0025570X.2020.1682447 To link to this article: https://doi.org/10.1080/0025570X.2020.1682447 Published online: 22 Jan 2020. Submit your article to this journal View related articles View Crossmark data Full Terms & Conditions of access and use can be found at https://www.tandfonline.com/action/journalInformation?journalCode=umma20 VOL. 93, NO. 1, FEBRUARY 2020 27 Efficiently Constructing Tangent Circles ARTHUR BARAGAR University of Nevada Las Vegas Las Vegas, NV 89154 [email protected] ALEX KONTOROVICH Rutgers University New Brunswick, NJ 08854 [email protected] The Greek geometers of antiquity devised a game—we might call it geometrical solitaire—which ...mustsurelystand at the very top of any list of games to be played alone. Over the ages it has attracted hosts of players, and though now well over 2000 years old, it seems not to have lost any of its singular charm or appeal. –HowardEves The Problem of Apollonius is to construct a circle tangent to three given ones in a plane. The three circles may also be limits of circles, that is, points or lines; and “con- struct” means using a straightedge and compass. Apollonius’s own solution did not survive antiquity [8], and we only know of its existence through a “mathscinet review” by Pappus half a millennium later. -
Pythagorean Theorem
MAC-CPTM Situations Project Situation 36: Pythagorean Theorem Prepared at Pennsylvania State University Mid-Atlantic Center for Mathematics Teaching and Learning June 28, 2005—Patrick Sullivan 27 September 2009 – Kathy Heid, Maureen Grady, Shiv Karunakaran Prompt In both an Algebra I course and an Advanced Algebra course, students were given transparency cutouts of graph paper squares with side lengths from one unit to twenty- five units. Students were asked to create triangles whose sides had the side-lengths of three of the squares. Students began to notice the squares that would create right triangles and the relationship involving the area of those squares. A student asked, “Does this work for every right triangle?” Commentary The Pythagorean Theorem relates the squares of the lengths of the sides of a right triangle. The Law of Cosines establishes a more general relationship between the same quantities that holds for all triangles. Using algebra and geometry together we can prove the Pythagorean Theorem and the Law of Cosines. SIT36_090927.doc Page 1 of 11 Mathematical Focus 1 Visual inspection alone is insufficient for drawing mathematical conclusions The generalization drawn by the student is based on what the student observed using physical models. Although such observations are important for mathematical discovery they cannot replace mathematical proof. The diagram that follows is an example of a case in which the physical representation is illusory. The diagram shown below makes it appear that two right triangles with the same base and height have different areas. However, close examination reveals that neither of the figures is actually a right triangle.