Saccheri and Lambert Quadrilateral in Hyperbolic Geometry
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Angle Chasing
Angle Chasing Ray Li June 12, 2017 1 Facts you should know 1. Let ABC be a triangle and extend BC past C to D: Show that \ACD = \BAC + \ABC: 2. Let ABC be a triangle with \C = 90: Show that the circumcenter is the midpoint of AB: 3. Let ABC be a triangle with orthocenter H and feet of the altitudes D; E; F . Prove that H is the incenter of 4DEF . 4. Let ABC be a triangle with orthocenter H and feet of the altitudes D; E; F . Prove (i) that A; E; F; H lie on a circle diameter AH and (ii) that B; E; F; C lie on a circle with diameter BC. 5. Let ABC be a triangle with circumcenter O and orthocenter H: Show that \BAH = \CAO: 6. Let ABC be a triangle with circumcenter O and orthocenter H and let AH and AO meet the circumcircle at D and E, respectively. Show (i) that H and D are symmetric with respect to BC; and (ii) that H and E are symmetric with respect to the midpoint BC: 7. Let ABC be a triangle with altitudes AD; BE; and CF: Let M be the midpoint of side BC. Show that ME and MF are tangent to the circumcircle of AEF: 8. Let ABC be a triangle with incenter I, A-excenter Ia, and D the midpoint of arc BC not containing A on the circumcircle. Show that DI = DIa = DB = DC: 9. Let ABC be a triangle with incenter I and D the midpoint of arc BC not containing A on the circumcircle. -
Point of Concurrency the Three Perpendicular Bisectors of a Triangle Intersect at a Single Point
3.1 Special Segments and Centers of Classifications of Triangles: Triangles By Side: 1. Equilateral: A triangle with three I CAN... congruent sides. Define and recognize perpendicular bisectors, angle bisectors, medians, 2. Isosceles: A triangle with at least and altitudes. two congruent sides. 3. Scalene: A triangle with three sides Define and recognize points of having different lengths. (no sides are concurrency. congruent) Jul 249:36 AM Jul 249:36 AM Classifications of Triangles: Special Segments and Centers in Triangles By angle A Perpendicular Bisector is a segment or line 1. Acute: A triangle with three acute that passes through the midpoint of a side and angles. is perpendicular to that side. 2. Obtuse: A triangle with one obtuse angle. 3. Right: A triangle with one right angle 4. Equiangular: A triangle with three congruent angles Jul 249:36 AM Jul 249:36 AM Point of Concurrency The three perpendicular bisectors of a triangle intersect at a single point. Two lines intersect at a point. The point of When three or more lines intersect at the concurrency of the same point, it is called a "Point of perpendicular Concurrency." bisectors is called the circumcenter. Jul 249:36 AM Jul 249:36 AM 1 Circumcenter Properties An angle bisector is a segment that divides 1. The circumcenter is an angle into two congruent angles. the center of the circumscribed circle. BD is an angle bisector. 2. The circumcenter is equidistant to each of the triangles vertices. m∠ABD= m∠DBC Jul 249:36 AM Jul 249:36 AM The three angle bisectors of a triangle Incenter properties intersect at a single point. -
Median and Altitude of a Triangle Goal: • to Use Properties of the Medians
Median and Altitude of a Triangle Goal: • To use properties of the medians of a triangle. • To use properties of the altitudes of a triangle. Median of a Triangle Median of a Triangle – a segment whose endpoints are the vertex of a triangle and the midpoint of the opposite side. Vertex Median Median of an Obtuse Triangle A D Point of concurrency “P” or centroid E P C B F Medians of a Triangle The medians of a triangle intersect at a point that is two-thirds of the distance from each vertex to the midpoint of the opposite side. A D If P is the centroid of ABC, then AP=2 AF E 3 P C CP=22CE and BP= BD B F 33 Example - Medians of a Triangle P is the centroid of ABC. PF 5 Find AF and AP A D E P C B F 5 Median of an Acute Triangle A Point of concurrency “P” or centroid E D P C B F Median of a Right Triangle A F Point of concurrency “P” or centroid E P C B D The three medians of an obtuse, acute, and a right triangle always meet inside the triangle. Altitude of a Triangle Altitude of a triangle – the perpendicular segment from the vertex to the opposite side or to the line that contains the opposite side A altitude C B Altitude of an Acute Triangle A Point of concurrency “P” or orthocenter P C B The point of concurrency called the orthocenter lies inside the triangle. Altitude of a Right Triangle The two legs are the altitudes A The point of concurrency called the orthocenter lies on the triangle. -
Special Isocubics in the Triangle Plane
Special Isocubics in the Triangle Plane Jean-Pierre Ehrmann and Bernard Gibert June 19, 2015 Special Isocubics in the Triangle Plane This paper is organized into five main parts : a reminder of poles and polars with respect to a cubic. • a study on central, oblique, axial isocubics i.e. invariant under a central, oblique, • axial (orthogonal) symmetry followed by a generalization with harmonic homolo- gies. a study on circular isocubics i.e. cubics passing through the circular points at • infinity. a study on equilateral isocubics i.e. cubics denoted 60 with three real distinct • K asymptotes making 60◦ angles with one another. a study on conico-pivotal isocubics i.e. such that the line through two isoconjugate • points envelopes a conic. A number of practical constructions is provided and many examples of “unusual” cubics appear. Most of these cubics (and many other) can be seen on the web-site : http://bernard.gibert.pagesperso-orange.fr where they are detailed and referenced under a catalogue number of the form Knnn. We sincerely thank Edward Brisse, Fred Lang, Wilson Stothers and Paul Yiu for their friendly support and help. Chapter 1 Preliminaries and definitions 1.1 Notations We will denote by the cubic curve with barycentric equation • K F (x,y,z) = 0 where F is a third degree homogeneous polynomial in x,y,z. Its partial derivatives ∂F ∂2F will be noted F ′ for and F ′′ for when no confusion is possible. x ∂x xy ∂x∂y Any cubic with three real distinct asymptotes making 60◦ angles with one another • will be called an equilateral cubic or a 60. -
Downloaded from Bookstore.Ams.Org 30-60-90 Triangle, 190, 233 36-72
Index 30-60-90 triangle, 190, 233 intersects interior of a side, 144 36-72-72 triangle, 226 to the base of an isosceles triangle, 145 360 theorem, 96, 97 to the hypotenuse, 144 45-45-90 triangle, 190, 233 to the longest side, 144 60-60-60 triangle, 189 Amtrak model, 29 and (logical conjunction), 385 AA congruence theorem for asymptotic angle, 83 triangles, 353 acute, 88 AA similarity theorem, 216 included between two sides, 104 AAA congruence theorem in hyperbolic inscribed in a semicircle, 257 geometry, 338 inscribed in an arc, 257 AAA construction theorem, 191 obtuse, 88 AAASA congruence, 197, 354 of a polygon, 156 AAS congruence theorem, 119 of a triangle, 103 AASAS congruence, 179 of an asymptotic triangle, 351 ABCD property of rigid motions, 441 on a side of a line, 149 absolute value, 434 opposite a side, 104 acute angle, 88 proper, 84 acute triangle, 105 right, 88 adapted coordinate function, 72 straight, 84 adjacency lemma, 98 zero, 84 adjacent angles, 90, 91 angle addition theorem, 90 adjacent edges of a polygon, 156 angle bisector, 100, 147 adjacent interior angle, 113 angle bisector concurrence theorem, 268 admissible decomposition, 201 angle bisector proportion theorem, 219 algebraic number, 317 angle bisector theorem, 147 all-or-nothing theorem, 333 converse, 149 alternate interior angles, 150 angle construction theorem, 88 alternate interior angles postulate, 323 angle criterion for convexity, 160 alternate interior angles theorem, 150 angle measure, 54, 85 converse, 185, 323 between two lines, 357 altitude concurrence theorem, -
Application of Nine Point Circle Theorem
Application of Nine Point Circle Theorem Submitted by S3 PLMGS(S) Students Bianca Diva Katrina Arifin Kezia Aprilia Sanjaya Paya Lebar Methodist Girls’ School (Secondary) A project presented to the Singapore Mathematical Project Festival 2019 Singapore Mathematic Project Festival 2019 Application of the Nine-Point Circle Abstract In mathematics geometry, a nine-point circle is a circle that can be constructed from any given triangle, which passes through nine significant concyclic points defined from the triangle. These nine points come from the midpoint of each side of the triangle, the foot of each altitude, and the midpoint of the line segment from each vertex of the triangle to the orthocentre, the point where the three altitudes intersect. In this project we carried out last year, we tried to construct nine-point circles from triangulated areas of an n-sided polygon (which we call the “Original Polygon) and create another polygon by connecting the centres of the nine-point circle (which we call the “Image Polygon”). From this, we were able to find the area ratio between the areas of the original polygon and the image polygon. Two equations were found after we collected area ratios from various n-sided regular and irregular polygons. Paya Lebar Methodist Girls’ School (Secondary) 1 Singapore Mathematic Project Festival 2019 Application of the Nine-Point Circle Acknowledgement The students involved in this project would like to thank the school for the opportunity to participate in this competition. They would like to express their gratitude to the project supervisor, Ms Kok Lai Fong for her guidance in the course of preparing this project. -
3. Adam Also Needs to Know the Altitude of the Smaller Triangle Within the Sign
Lesson 3: Special Right Triangles Standard: MCC9‐12.G.SRT.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Essential Question: What are the relationships among the side lengths of a 30° - 60° - 90° triangle? How can I use these relationships to solve real-world problems? Discovering Special Triangles Learning Task (30-60-90 triangle) 1. Adam, a construction manager in a nearby town, needs to check the uniformity of Yield signs around the state and is checking the heights (altitudes) of the Yield signs in your locale. Adam knows that all yield signs have the shape of an equilateral triangle. Why is it sufficient for him to check just the heights (altitudes) of the signs to verify uniformity? 2. A Yield sign created in Adam’s plant is pictured above. It has the shape of an equilateral triangle with a side length of 2 feet. If you draw the altitude of the triangular sign, you split the Yield sign in half vertically. Use what you know about triangles to find the angle measures and the side lengths of the two smaller triangles formed by the altitude. a. What does an altitude of an equilateral triangle do to the triangle? b. What kinds of triangles are formed when you draw the altitude? c. What did the altitude so to the base of the triangle? d. What did the altitude do to the angle from which it was drawn? 3. Adam also needs to know the altitude of the smaller triangle within the sign. -
Geometry: Neutral MATH 3120, Spring 2016 Many Theorems of Geometry Are True Regardless of Which Parallel Postulate Is Used
Geometry: Neutral MATH 3120, Spring 2016 Many theorems of geometry are true regardless of which parallel postulate is used. A neutral geom- etry is one in which no parallel postulate exists, and the theorems of a netural geometry are true for Euclidean and (most) non-Euclidean geomteries. Spherical geometry is a special case of Non-Euclidean geometries where the great circles on the sphere are lines. This leads to spherical trigonometry where triangles have angle measure sums greater than 180◦. While this is a non-Euclidean geometry, spherical geometry develops along a separate path where the axioms and theorems of neutral geometry do not typically apply. The axioms and theorems of netural geometry apply to Euclidean and hyperbolic geometries. The theorems below can be proven using the SMSG axioms 1 through 15. In the SMSG axiom list, Axiom 16 is the Euclidean parallel postulate. A neutral geometry assumes only the first 15 axioms of the SMSG set. Notes on notation: The SMSG axioms refer to the length or measure of line segments and the measure of angles. Thus, we will use the notation AB to describe a line segment and AB to denote its length −−! −! or measure. We refer to the angle formed by AB and AC as \BAC (with vertex A) and denote its measure as m\BAC. 1 Lines and Angles Definitions: Congruence • Segments and Angles. Two segments (or angles) are congruent if and only if their measures are equal. • Polygons. Two polygons are congruent if and only if there exists a one-to-one correspondence between their vertices such that all their corresponding sides (line sgements) and all their corre- sponding angles are congruent. -
The Algebra of Projective Spheres on Plane, Sphere and Hemisphere
Journal of Applied Mathematics and Physics, 2020, 8, 2286-2333 https://www.scirp.org/journal/jamp ISSN Online: 2327-4379 ISSN Print: 2327-4352 The Algebra of Projective Spheres on Plane, Sphere and Hemisphere István Lénárt Eötvös Loránd University, Budapest, Hungary How to cite this paper: Lénárt, I. (2020) Abstract The Algebra of Projective Spheres on Plane, Sphere and Hemisphere. Journal of Applied Numerous authors studied polarities in incidence structures or algebrization Mathematics and Physics, 8, 2286-2333. of projective geometry [1] [2]. The purpose of the present work is to establish https://doi.org/10.4236/jamp.2020.810171 an algebraic system based on elementary concepts of spherical geometry, ex- tended to hyperbolic and plane geometry. The guiding principle is: “The Received: July 17, 2020 Accepted: October 27, 2020 point and the straight line are one and the same”. Points and straight lines are Published: October 30, 2020 not treated as dual elements in two separate sets, but identical elements with- in a single set endowed with a binary operation and appropriate axioms. It Copyright © 2020 by author(s) and consists of three sections. In Section 1 I build an algebraic system based on Scientific Research Publishing Inc. This work is licensed under the Creative spherical constructions with two axioms: ab= ba and (ab)( ac) = a , pro- Commons Attribution International viding finite and infinite models and proving classical theorems that are License (CC BY 4.0). adapted to the new system. In Section Two I arrange hyperbolic points and http://creativecommons.org/licenses/by/4.0/ straight lines into a model of a projective sphere, show the connection be- Open Access tween the spherical Napier pentagram and the hyperbolic Napier pentagon, and describe new synthetic and trigonometric findings between spherical and hyperbolic geometry. -
The Role of Definitions in Geometry for Prospective Middle and High School Teachers
2018 HAWAII UNIVERSITY INTERNATIONAL CONFERENCES STEAM - SCIENCE, TECHNOLOGY & ENGINEERING, ARTS, MATHEMATICS & EDUCATION JUNE 6 - 8, 2018 PRINCE WAIKIKI, HONOLULU, HAWAII THE ROLE OF DEFINITIONS IN GEOMETRY FOR PROSPECTIVE MIDDLE AND HIGH SCHOOL TEACHERS SZABO, TAMAS DEPARTMENT OF MATHEMATICS UNIVERSITY OF WISCONSIN WHITEWATER, WISCONSIN Dr. Tamas Szabo Department of Mathematics University of Wisconsin Whitewater, Wisconsin THE ROLE OF DEFINITIONS IN GEOMETRY FOR PROSPECTIVE MIDDLE AND HIGH SCHOOL TEACHERS ABSTRACT. Does verbalizing a key definition before solving a related problem help students solving the problem? Perhaps surprisingly, the answers turns out to be NO, based on this yearlong experiment conducted with mathematics education majors and minors in two geometry classes. This article describes the experiment, the results, suggests possible explanations, and derives some peda-gogical conclusions that are useful for teachers of any mathematics or science course. Keywords: concept definition, concept image, proof writing, problem solving, secondary mathematics teachers. 1. INTRODUCTION Definitions play a very important role in mathematics, the learning of mathematics, and the teaching of mathematics alike. However, knowing a definition is not equivalent with knowing a concept. To become effective problem solvers, students need to develop an accurate concept image. The formal definition is only a small part of the concept image, which also includes examples and non-examples, properties and connections to other concepts. The distinction between concept definition and concept image has been examined by many research studies (e.g., Edwards, 1997; Vinner and Dreyfus, 1989; Tall, 1992; Vinner, 1991). Another group of articles analyze, argue for, and provide examples of engaging students in the construction of definitions (e.g., Zandieh and Rasmussen, 2010; Herbst et al., 2005; Johnson et al., 2014; Zaslavski and Shir, 2005). -
The Euler Line in Non-Euclidean Geometry
California State University, San Bernardino CSUSB ScholarWorks Theses Digitization Project John M. Pfau Library 2003 The Euler Line in non-Euclidean geometry Elena Strzheletska Follow this and additional works at: https://scholarworks.lib.csusb.edu/etd-project Part of the Mathematics Commons Recommended Citation Strzheletska, Elena, "The Euler Line in non-Euclidean geometry" (2003). Theses Digitization Project. 2443. https://scholarworks.lib.csusb.edu/etd-project/2443 This Thesis is brought to you for free and open access by the John M. Pfau Library at CSUSB ScholarWorks. It has been accepted for inclusion in Theses Digitization Project by an authorized administrator of CSUSB ScholarWorks. For more information, please contact [email protected]. THE EULER LINE IN NON-EUCLIDEAN GEOMETRY A Thesis Presented to the Faculty of California State University, San Bernardino In Partial Fulfillment of the Requirements for the Degree Master of Arts in Mathematics by Elena Strzheletska December 2003 THE EULER LINE IN NON-EUCLIDEAN GEOMETRY A Thesis Presented to the Faculty of California State University, San Bernardino by Elena Strzheletska December 2003 Approved by: Robert Stein, Committee Member Susan Addington, Committee Member Peter Williams, Chair Terry Hallett, Department of Mathematics Graduate Coordinator Department of Mathematics ABSTRACT In Euclidean geometry, the circumcenter and the centroid of a nonequilateral triangle determine a line called the Euler line. The orthocenter of the triangle, the point of intersection of the altitudes, also belongs to this line. The main purpose of this thesis is to explore the conditions of the existence and the properties of the Euler line of a triangle in the hyperbolic plane. -
Volume 6 (2006) 1–16
FORUM GEOMETRICORUM A Journal on Classical Euclidean Geometry and Related Areas published by Department of Mathematical Sciences Florida Atlantic University b bbb FORUM GEOM Volume 6 2006 http://forumgeom.fau.edu ISSN 1534-1178 Editorial Board Advisors: John H. Conway Princeton, New Jersey, USA Julio Gonzalez Cabillon Montevideo, Uruguay Richard Guy Calgary, Alberta, Canada Clark Kimberling Evansville, Indiana, USA Kee Yuen Lam Vancouver, British Columbia, Canada Tsit Yuen Lam Berkeley, California, USA Fred Richman Boca Raton, Florida, USA Editor-in-chief: Paul Yiu Boca Raton, Florida, USA Editors: Clayton Dodge Orono, Maine, USA Roland Eddy St. John’s, Newfoundland, Canada Jean-Pierre Ehrmann Paris, France Chris Fisher Regina, Saskatchewan, Canada Rudolf Fritsch Munich, Germany Bernard Gibert St Etiene, France Antreas P. Hatzipolakis Athens, Greece Michael Lambrou Crete, Greece Floor van Lamoen Goes, Netherlands Fred Pui Fai Leung Singapore, Singapore Daniel B. Shapiro Columbus, Ohio, USA Steve Sigur Atlanta, Georgia, USA Man Keung Siu Hong Kong, China Peter Woo La Mirada, California, USA Technical Editors: Yuandan Lin Boca Raton, Florida, USA Aaron Meyerowitz Boca Raton, Florida, USA Xiao-Dong Zhang Boca Raton, Florida, USA Consultants: Frederick Hoffman Boca Raton, Floirda, USA Stephen Locke Boca Raton, Florida, USA Heinrich Niederhausen Boca Raton, Florida, USA Table of Contents Khoa Lu Nguyen and Juan Carlos Salazar, On the mixtilinear incircles and excircles,1 Juan Rodr´ıguez, Paula Manuel and Paulo Semi˜ao, A conic associated with the Euler line,17 Charles Thas, A note on the Droz-Farny theorem,25 Paris Pamfilos, The cyclic complex of a cyclic quadrilateral,29 Bernard Gibert, Isocubics with concurrent normals,47 Mowaffaq Hajja and Margarita Spirova, A characterization of the centroid using June Lester’s shape function,53 Christopher J.