The Tangent Circle Problem

Ryan Hirst

Tackling NatPBS' tangent circles: [url]http://www.geogebratube.org/material/show/id/43442[/url] The aim of this collection is to solve a difficult problem of elementary mathematics, from scratch.

Table of Contents

  1. The Arbelos and Pappus' Chain
    1. Archimedes' Arbelos: 1
    2. Arbelos 1c: Vector Reduction
    3. Ellipse by parameter, scale, rotation
    4. Arbelos 1e: Final Diagram
  2. Steiner's Chain
    1. Arbelos, 2a
    2. Arbelos 2b
    3. Ring of tangent Circles
    4. Arbelos Chain, Final
  3. Iterating into the Gaps
    1. Tangent Chain to equal Circles
    2. Equal circles 2, Initial position
    3. Four Mutually Tangent & Exterior Circles
    4. Mutually Exterior & Tangent Circles, 2
    5. Translation, Scale, Rotation
    6. Reflection: Line about a Circle
    7. Reflection: Circle about a Circle
    8. Circle Reflection: Metric Space
    9. Circles on the Rim, 2
    10. Circles on the Rim, 3: Iteration
    11. Tangent Circle Problem: Solution 1

1.

The Arbelos and Pappus' Chain

  • 1. Archimedes' Arbelos: 1
  • 2. Arbelos 1c: Vector Reduction
  • 3. Ellipse by parameter, scale, rotation
  • 4. Arbelos 1e: Final Diagram

1.1.

Archimedes' Arbelos: 1

Subproblem: To inscribe a circle in the knife.

_____________________ Archimedes' Arbelos: [list] [*][b] →1a. Inscribe a circle in the arc.[/b] [*]1b. Tangent circles in the arc (Solution 1). [*]1c. Vector Reduction: [url]http://www.geogebratube.org/material/show/id/54557[/url] [*]1d. Ellipse from one parameter, scale and rotation: [url]http://www.geogebratube.org/material/show/id/55256[/url] [*]1e. Final Construction: [url]http://www.geogebratube.org/material/show/id/54592[/url] [*]2a. Let one circle enclose another. Inscribe a third circle in the ring: [url]http://www.geogebratube.org/material/show/id/54595[/url] [*]2b. Tangent circles in the ring. [url]http://www.geogebratube.org/material/show/id/54596[/url] [/list] 3. Cyclic Solution: [list] [*]3a. An outer ring of tangent circles: [url]http://www.geogebratube.org/material/show/id/55009[/url] [*]3b. Determine the projection. [*]3c. Final Construction: [url]http://www.geogebratube.org/material/show/id/55883[/url] [/list]
– Ryan Hirst

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2.

Steiner's Chain

  • 1. Arbelos, 2a
  • 2. Arbelos 2b
  • 3. Ring of tangent Circles
  • 4. Arbelos Chain, Final

2.1.

Arbelos, 2a

For a single circle, the procedure is identical. ...

_____________________ Archimedes' Arbelos: [list] [*]1a. Inscribe a circle in the arc.[url]http://www.geogebratube.org/material/show/id/54105[/url] [*]1b. Tangent circles in the arc (Solution 1). [*]1c. Vector Reduction: [url]http://www.geogebratube.org/material/show/id/54557[/url] [*]1d. Ellipse from parameter, scale and rotation:[url]http://www.geogebratube.org/material/show/id/55256[/url] [*]1e. Final Construction: [url]http://www.geogebratube.org/material/show/id/54592[/url] [*][b]→2a. Let one circle enclose another. Inscribe a third circle in the ring. [/b] [*]2b. Tangent circles in the ring. [url]http://www.geogebratube.org/material/show/id/54596[/url] [/list] 3. Cyclic Solution: [list] [*]3a. An outer ring of tangent circles: [url]http://www.geogebratube.org/material/show/id/55009[/url] [*]3b. Determine the projection. [*]3c. Final Construction: [url]http://www.geogebratube.org/material/show/id/55883[/url] [/list]
– Ryan Hirst

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3.

Iterating into the Gaps

  • 1. Tangent Chain to equal Circles
  • 2. Equal circles 2, Initial position
  • 3. Four Mutually Tangent & Exterior Circles
  • 4. Mutually Exterior & Tangent Circles, 2
  • 5. Translation, Scale, Rotation
  • 6. Reflection: Line about a Circle
  • 7. Reflection: Circle about a Circle
  • 8. Circle Reflection: Metric Space
  • 9. Circles on the Rim, 2
  • 10. Circles on the Rim, 3: Iteration
  • 11. Tangent Circle Problem: Solution 1

3.1.

Tangent Chain to equal Circles

If I have a starting circle, where is the next one?

The radius of the [i]n[/i]th circle is [math] x_n [/math], and its center is a distance [math] r k_n[/math] from [math]M [/math]. The sequence works for any starting position. _________________ The Tangent Circle Problem: [list] [*]1. Tangent along the rim: solve for k [*]2a. Initial position: [url]http://www.geogebratube.org/material/show/id/58360[/url] [*][b]→ 2b. Tangent chain to equal circles[/b] [*]3a. Four mutually tangent & exterior circles (Apollonius): [url]http://www.geogebratube.org/material/show/id/58189 [/url] [*]3b. Vector reduction: [url]http://www.geogebratube.org/material/show/id/58461[/url] [/list] [list] [*]Affine Transformation [url]http://www.geogebratube.org/material/show/id/58177[/url] [*]Reflection: Line about a Circle [url]http://www.geogebratube.org/material/show/id/58522[/url] [*]Reflection: Circle about a Circle: [url]http://www.geogebratube.org/material/show/id/58185[/url] [*]Circle Inversion: Metric Space: [url]http://www.geogebratube.org/material/show/id/60132[/url] [/list] Solution: [list] [*]Sequences 1: Formation [url]http://www.geogebratube.org/material/show/id/58896[/url] [*]Sequence 1: Formation [url]http://www.geogebratube.org/material/show/id/59816[/url] [*]Sequence 1: Iteration 1 [url]http://www.geogebratube.org/material/show/id/59828[/url] [*]Example of equivalent projections: [url]http://www.geogebratube.org/material/show/id/65754[/url] [*]Final Diagram: [url]http://www.geogebratube.org/material/show/id/65755[/url] [/list]
– Ryan Hirst

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3.11.

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