• Study Resource
  • Explore
    • Arts & Humanities
    • Business
    • Engineering & Technology
    • Foreign Language
    • History
    • Math
    • Science
    • Social Science

    Top subcategories

    • Advanced Math
    • Algebra
    • Basic Math
    • Calculus
    • Geometry
    • Linear Algebra
    • Pre-Algebra
    • Pre-Calculus
    • Statistics And Probability
    • Trigonometry
    • other →

    Top subcategories

    • Astronomy
    • Astrophysics
    • Biology
    • Chemistry
    • Earth Science
    • Environmental Science
    • Health Science
    • Physics
    • other →

    Top subcategories

    • Anthropology
    • Law
    • Political Science
    • Psychology
    • Sociology
    • other →

    Top subcategories

    • Accounting
    • Economics
    • Finance
    • Management
    • other →

    Top subcategories

    • Aerospace Engineering
    • Bioengineering
    • Chemical Engineering
    • Civil Engineering
    • Computer Science
    • Electrical Engineering
    • Industrial Engineering
    • Mechanical Engineering
    • Web Design
    • other →

    Top subcategories

    • Architecture
    • Communications
    • English
    • Gender Studies
    • Music
    • Performing Arts
    • Philosophy
    • Religious Studies
    • Writing
    • other →

    Top subcategories

    • Ancient History
    • European History
    • US History
    • World History
    • other →

    Top subcategories

    • Croatian
    • Czech
    • Finnish
    • Greek
    • Hindi
    • Japanese
    • Korean
    • Persian
    • Swedish
    • Turkish
    • other →
 
Profile Documents Logout
Upload
Unit 9 Vocabulary and Objectives File
Unit 9 Vocabulary and Objectives File

Unit 3 | Circles and Volume
Unit 3 | Circles and Volume

Study Guide - U.I.U.C. Math
Study Guide - U.I.U.C. Math

Geometry Chapter 12 Quiz Review Assume the lines that look like
Geometry Chapter 12 Quiz Review Assume the lines that look like

... Determine if the following are tangents. ...
G.9 - DPS ARE
G.9 - DPS ARE

G.9 - DPS ARE
G.9 - DPS ARE

Full text
Full text

Geometry §14.6 Angles formed by Tangents: Tangent: A line that
Geometry §14.6 Angles formed by Tangents: Tangent: A line that

Geometry Level 1 Curriculum
Geometry Level 1 Curriculum

... of a circle 3. If a radius is perp to a chord, then it bisects the chord (and the converse) 4. The perp. bisector of a chord passes through the center of the circle 5. Vocabulary: sector, circle, center, radius, concentric, interior, exterior, diameter, chord, arc, central angle, minor arc, major ar ...
CP Geometry Circles Unit 12 Syllabus
CP Geometry Circles Unit 12 Syllabus

NCDJJDP Lesson Plan
NCDJJDP Lesson Plan

... How do you use the properties of tangents and chords to find missing measures in circles? How do you use central angles to find arc measures? How do you use inscribed angles to find missing measures in circles? How do you use circumference to find arclength? ...
Geometry – Chapter 1
Geometry – Chapter 1

... 10.2 Use properties of arcs and chords of a circle 10.3 Use inscribed angles and inscribed polygons 10.4 Use angles formed by tangents and chords 10.4 Use angles formed by lines that intersect a circle 10.5 Find the lengths of segments of chords ...
Conics - Circles
Conics - Circles

... Circles: Notice that to create a circle by intersecting a cone and a plane, the plane must be parallel to the base of the cone. Any of these conics can be graphed on a coordinate plane. The graph of any conic on an x-y coordinate plane can be described by an equation in the form: Ax2 + Bxy + Cy2 + D ...
6.1 Introduction to Circle Notes
6.1 Introduction to Circle Notes

... 4. A _______________________ of a circle is a chord that passes through the center of the circle. 5. A _______________________ is a line that intersects a circle at two points. A secant of a circle includes a chord of the circle. 6. A _______________________ of a circle is a line that intersects a c ...
Unit 6 Lesson 7 Outline
Unit 6 Lesson 7 Outline

... Lesson Plan Outline Geometry in Construction Title: Tangent Lines to Circles ...
Geometry Level 3 Curriculum
Geometry Level 3 Curriculum

Geometry Worksheet: Angles formed by Chords, Secants, and
Geometry Worksheet: Angles formed by Chords, Secants, and

Name
Name

Formulas
Formulas

Geometry
Geometry

...  If a line is tangent to a circle, then the line is  to the radius drawn to the point of tangency.  If a line is  to a radius at its outer endpoint, then the line is tangent to the circle.  Minor arcs are  if and only if their central angles are  .  In the same circle or in  circles: (1)  ...
“I can” Objective - Liberty Union High School District
“I can” Objective - Liberty Union High School District

Geometry – Chapter 1
Geometry – Chapter 1

Geometry: Circles Name: ____TEACHER COPY CCSS.Math
Geometry: Circles Name: ____TEACHER COPY CCSS.Math

... Investigation (23 minutes): Students will be working with partners but on their own paper. Each student should have an example for each problem on their own paper. While students work in their groups, the teacher will circulate around the classroom and make clarifications and corrections as necessar ...
Geometry: Circles Name: ____TEACHER COPY CCSS.Math
Geometry: Circles Name: ____TEACHER COPY CCSS.Math

... Investigation (23 minutes): Students will be working with partners but on their own paper. Each student should have an example for each problem on their own paper. While students work in their groups, the teacher will circulate around the classroom and make clarifications and corrections as necessar ...
NAME - Livingston Public Schools
NAME - Livingston Public Schools

< 1 ... 41 42 43 44 45 >

Problem of Apollonius



In Euclidean plane geometry, Apollonius's problem is to construct circles that are tangent to three given circles in a plane (Figure 1). Apollonius of Perga (ca. 262 BC – ca. 190 BC) posed and solved this famous problem in his work Ἐπαφαί (Epaphaí, ""Tangencies""); this work has been lost, but a 4th-century report of his results by Pappus of Alexandria has survived. Three given circles generically have eight different circles that are tangent to them (Figure 2) and each solution circle encloses or excludes the three given circles in a different way: in each solution, a different subset of the three circles is enclosed (its complement is excluded) and there are 8 subsets of a set whose cardinality is 3, since 8 = 23.In the 16th century, Adriaan van Roomen solved the problem using intersecting hyperbolas, but this solution does not use only straightedge and compass constructions. François Viète found such a solution by exploiting limiting cases: any of the three given circles can be shrunk to zero radius (a point) or expanded to infinite radius (a line). Viète's approach, which uses simpler limiting cases to solve more complicated ones, is considered a plausible reconstruction of Apollonius' method. The method of van Roomen was simplified by Isaac Newton, who showed that Apollonius' problem is equivalent to finding a position from the differences of its distances to three known points. This has applications in navigation and positioning systems such as LORAN.Later mathematicians introduced algebraic methods, which transform a geometric problem into algebraic equations. These methods were simplified by exploiting symmetries inherent in the problem of Apollonius: for instance solution circles generically occur in pairs, with one solution enclosing the given circles that the other excludes (Figure 2). Joseph Diaz Gergonne used this symmetry to provide an elegant straightedge and compass solution, while other mathematicians used geometrical transformations such as reflection in a circle to simplify the configuration of the given circles. These developments provide a geometrical setting for algebraic methods (using Lie sphere geometry) and a classification of solutions according to 33 essentially different configurations of the given circles.Apollonius' problem has stimulated much further work. Generalizations to three dimensions—constructing a sphere tangent to four given spheres—and beyond have been studied. The configuration of three mutually tangent circles has received particular attention. René Descartes gave a formula relating the radii of the solution circles and the given circles, now known as Descartes' theorem. Solving Apollonius' problem iteratively in this case leads to the Apollonian gasket, which is one of the earliest fractals to be described in print, and is important in number theory via Ford circles and the Hardy–Littlewood circle method.
  • studyres.com © 2025
  • DMCA
  • Privacy
  • Terms
  • Report