Circles (Theorems)

Tim Brzezinski, Sep 17, 2015

Circle Theorems, Circle Properties, etc...

Table of Contents

  1. Circle Vocabulary
    1. Circle Terminology
    2. Circle Tangent-Types Illustrator
    3. Similar Circles?
    4. Congruent Circles: Definition
  2. Tangents
    1. Properties of Tangents Drawn to Circles (A)
    2. Properties of Tangents Drawn to Circles (B)
    3. Properties of Tangents
    4. Tangents to Circles: Investigation
    5. Int+Ext Tangent Action!
    6. GoGeometry Action 6!
  3. Inscribed Angle Theorems & Corollaries
    1. Inscribed Angle Theorem (V1)
    2. Inscribed Angle Theorem: Take 2!
    3. Inscribed Angle Theorems: Take 3!
    4. Inscribed Angle Theorem: Take 4!
    5. Proof Exercise: Inscribed Angle Theorem (Case 1)
    6. Proof Exercise: Inscribed Angle Theorem (Case 2)
    7. Proof Exercise: Inscribed Angle Theorem (Case 3)
    8. Where to Sit? (I)
    9. Where to Sit (II)?
    10. Inscribed Angle Theorem: Corollary 1
    11. Inscribed Angle Theorems: Take 4!
    12. Inscribed Angle Theorem Dance: Take 2!
    13. Animation 20 (Inscribed Angle Dance!)
    14. Thales' Theorem (VA)
    15. Thales' Theorem (VB)
    16. Inscribed Angle Intercepts Semicircle
    17. Cyclic Quadrilaterals (IAT: Corollary 3)
    18. Cyclic Quadrilateral: Proof Hint
    19. Geometric Mean Illustration
  4. Theorems Involving Chords
    1. Congruent Chords: Quick Investigation
    2. Congruent Chords (I)
    3. Diameters & Chords I (VA)
    4. Diameters & Chords I (VB)
    5. Angle Formed by 2 Chords (I)
    6. Angle Formed by 2 Chords (II)
    7. Angle Formed by Chord & Tangent (I)
    8. Angle Formed by Chord & Tangent (II)
    9. Crossing Chords: Proof Hint
    10. Congruent Chords Action (A)
    11. Equidistant Chord Action
    12. Butterfly Theorem Action!
    13. Cut-The-Knot Action (2) !!!
    14. GoGeometry Action 1!
    15. GoGeometry Action 2!
    16. GoGeometry Action 3!
    17. GoGeometry Action 5!
    18. GoGeometry Action 8!
    19. Cut-The-Knot Action (6)!
    20. GoGeometry Action 9!
    21. GoGeometry Action 15!
    22. Cut-The-Knot Action 7!
    23. GoGeometry Action 19!
  5. Secant & Tangent Theorems
    1. Angles from Secants and Tangents (V1)
    2. Angle From 2 Secants (V2)
    3. Secants: Proof Hint
    4. Not Your Everyday Chord & Tangent Theorem
    5. GoGeometry Action 4!
    6. GoGeometry Action 13!
    7. GoGeometry Action 16!
    8. Thales Action + Sequel = GoGeometry Action 25!
    9. GoGeometry Action 26!
  6. Older (Earlier) Applets
    1. Inscribed Angle Theorem (Proof without Words)
    2. Inscribed Angle Theorem (Corollary 1) (Proof without Words)
    3. Inscribed Angle Theorem (Corollary 2) (Proof without Words)
    4. Inscribed Angle Theorem (Corollary 3) Cyclic Quadrilaterals
    5. Angle formed by a Secant & Tangent: What Do You See? (V1)
    6. Angle Formed by a Secant & Tangent: What Do You See? (V2)
    7. Angle Formed by 2 Tangents (V2): What Do You See?
    8. Cyclic Quadrilaterals
    9. Congruent Chords in the Same Circle
    10. Inscribed Angle Theorem (Intro): What Do You See Here?
    11. Inscribed Angle Action !!!
    12. Inscribed Angle Action! (2 Theorems)
    13. Angle Formed by 2 Chords (V1): What Do You See?
    14. Angle Formed by 2 Chords (V2): What Do You See?
    15. Chords Equidistant from Circle's Center
    16. Crossing Chords Property & Proof Start
    17. Crossing Chords Property & Proof Start (II)

1.

Circle Vocabulary

  • 1. Circle Terminology
  • 2. Circle Tangent-Types Illustrator
  • 3. Similar Circles?
  • 4. Congruent Circles: Definition

1.1.

Circle Terminology

[color=#000000]There are many vocabulary terms we use when talking about a circle. [br]The following app was designed to help you clearly see and interact with each term. [br][br]Explore this app for a few minutes. Then answer the questions that follow. [/color]
Note: LARGE POINTS are moveable.
How would you describe or define a [b]CIRCLE[/b] as a locus (set of points that meets specified criteria)?
How would you describe the term [b][color=#38761d]RADIUS[/color] [/b][i]without using the words "half" or "diameter" [/i]in your description?
What does the term [b][color=#9900ff]CHORD[/color] [/b]mean here in the context of a circle?
How would you describe the term [b][color=#ff7700]DIAMETER[/color] [/b][i]without using the words "two", "double", or "diameter" [/i]in your description?
How would you describe/define the term [b][color=#cc0000]SECANT[/color][/b]?
What does it mean for a line to be [b][color=#1e84cc]TANGENT [/color][/b]to a circle?
– Tim Brzezinski

1.2.

–

1.3.

–

1.4.

–

2.

Tangents

  • 1. Properties of Tangents Drawn to Circles (A)
  • 2. Properties of Tangents Drawn to Circles (B)
  • 3. Properties of Tangents
  • 4. Tangents to Circles: Investigation
  • 5. Int+Ext Tangent Action!
  • 6. GoGeometry Action 6!

2.1.

Properties of Tangents Drawn to Circles (A)

In the applet below, 2 tangent rays are drawn to a circle from a point outside that circle. [br][br]Interact with the applet below for a few minutes. As you do, be sure to change the locations of the [b][color=#3c78d8]blue points[/color][/b] (i.e. alter the size of the circle) and [b][color=#bf9000]yellow point[/color][/b]. [br] [br]After doing so, please answer the questions that follow.
1.
What is the measure of each [b][color=#ff00ff]pink angle[/color][/b] displayed? Explain how you know this.
2.
What can you conclude about the [b]distances[/b] from the [b][color=#bf9000]yellow point (outside the circle)[/color][/b] to each of the smaller white points on the circle?
Quick (Silent) Demo
3.
Given your response for (1), how could we prove that your response to (2) is true? (Assume we didn't see the rotation at the very end of the animation).
– Tim Brzezinski

2.2.

–

2.3.

–

2.4.

–

2.5.

–

2.6.

–

3.

Inscribed Angle Theorems & Corollaries

  • 1. Inscribed Angle Theorem (V1)
  • 2. Inscribed Angle Theorem: Take 2!
  • 3. Inscribed Angle Theorems: Take 3!
  • 4. Inscribed Angle Theorem: Take 4!
  • 5. Proof Exercise: Inscribed Angle Theorem (Case 1)
  • 6. Proof Exercise: Inscribed Angle Theorem (Case 2)
  • 7. Proof Exercise: Inscribed Angle Theorem (Case 3)
  • 8. Where to Sit? (I)
  • 9. Where to Sit (II)?
  • 10. Inscribed Angle Theorem: Corollary 1
  • 11. Inscribed Angle Theorems: Take 4!
  • 12. Inscribed Angle Theorem Dance: Take 2!
  • 13. Animation 20 (Inscribed Angle Dance!)
  • 14. Thales' Theorem (VA)
  • 15. Thales' Theorem (VB)
  • 16. Inscribed Angle Intercepts Semicircle
  • 17. Cyclic Quadrilaterals (IAT: Corollary 3)
  • 18. Cyclic Quadrilateral: Proof Hint
  • 19. Geometric Mean Illustration

3.1.

Inscribed Angle Theorem (V1)

[b][color=#ff00ff]The PINK ANGLE is said to be an INSCRIBED ANGLE[/color][/b] of a circle. [br][br]You can move the pink point anywhere on the NON-BLUE arc of the circle. [br][color=#0000ff][b]You can change the size of the BLUE intercepted arc[/b][/color] by moving either of the white points. [br]You can also adjust the circle's radius using the [color=#666666][b]GRAY POINT[/b][/color]. [br][br]Answer the questions that follow.
1.
Without looking up the definition on another tab in your internet browser, [b][color=#ff00ff]how would you describe (define) the concept of an inscribed angle of a circle? [/color][/b]
2.
[b][color=#ff00ff]How many inscribed angles[/color][/b] fit inside the [b][color=#0000ff]blue central angle[/color][/b] that intercepts (cuts off) the [b][color=#0000ff]same arc[/color][/b]?
3.
Given your result for (2), how does the [b][color=#ff00ff]measure of the pink inscribed angle[/color][/b] compare with the [color=#0000ff][b]measure of the blue intercepted arc? [/b][/color]
4.
Try testing your informal conclusions for (responses to) (2) and (3) a few times by dragging the slider back to its starting position, [b][color=#ff00ff]changing the location of the pink inscribed angle[/color][/b], and [b][color=#0000ff]changing the size of the blue intercepted arc[/color][/b]. [br][br]Then slide the slider again. [br][b][br]Do your conclusions for (2) and (3) ALWAYS hold true? [/b]
Quick (Silent) Demo
– Tim Brzezinski

3.2.

–

3.3.

–

3.4.

–

3.5.

–

3.6.

–

3.7.

–

3.8.

–

3.9.

–

3.10.

–

3.11.

–

3.12.

–

3.13.

–

3.14.

–

3.15.

–

3.16.

–

3.17.

–

3.18.

–

3.19.

–

4.

Theorems Involving Chords

  • 1. Congruent Chords: Quick Investigation
  • 2. Congruent Chords (I)
  • 3. Diameters & Chords I (VA)
  • 4. Diameters & Chords I (VB)
  • 5. Angle Formed by 2 Chords (I)
  • 6. Angle Formed by 2 Chords (II)
  • 7. Angle Formed by Chord & Tangent (I)
  • 8. Angle Formed by Chord & Tangent (II)
  • 9. Crossing Chords: Proof Hint
  • 10. Congruent Chords Action (A)
  • 11. Equidistant Chord Action
  • 12. Butterfly Theorem Action!
  • 13. Cut-The-Knot Action (2) !!!
  • 14. GoGeometry Action 1!
  • 15. GoGeometry Action 2!
  • 16. GoGeometry Action 3!
  • 17. GoGeometry Action 5!
  • 18. GoGeometry Action 8!
  • 19. Cut-The-Knot Action (6)!
  • 20. GoGeometry Action 9!
  • 21. GoGeometry Action 15!
  • 22. Cut-The-Knot Action 7!
  • 23. GoGeometry Action 19!

4.1.

Congruent Chords: Quick Investigation

Interact with the applet below for a minute or two. [br][br]If [b]two chords of a circle are congruent[/b], what can we conclude about the [b][color=#980000]arcs[/color][/b] these chords determine (i.e. "cut off")?
Quick Demo: 0:00 sec - 0:58 sec (BGM: Andy Hunter)
– Tim Brzezinski

4.2.

–

4.3.

–

4.4.

–

4.5.

–

4.6.

–

4.7.

–

4.8.

–

4.9.

–

4.10.

–

4.11.

–

4.12.

–

4.13.

–

4.14.

–

4.15.

–

4.16.

–

4.17.

–

4.18.

–

4.19.

–

4.20.

–

4.21.

–

4.22.

–

4.23.

–

5.

Secant & Tangent Theorems

  • 1. Angles from Secants and Tangents (V1)
  • 2. Angle From 2 Secants (V2)
  • 3. Secants: Proof Hint
  • 4. Not Your Everyday Chord & Tangent Theorem
  • 5. GoGeometry Action 4!
  • 6. GoGeometry Action 13!
  • 7. GoGeometry Action 16!
  • 8. Thales Action + Sequel = GoGeometry Action 25!
  • 9. GoGeometry Action 26!

5.1.

Angles from Secants and Tangents (V1)

[color=#000000]Interact with the applet below for a few minutes. Then answer the questions that follow. [br][br][/color][color=#0000ff]Be sure to change the locations of the[/color][color=#000000] [b]BIG POINTS[/b] [/color][color=#0000ff]each time [i]before[/i] you slide the slider. [/color]
1.
Suppose the [color=#ff00ff][b]entire pink arc measures 200 degrees[/b][/color] and the [color=#1e84cc][b]entire blue arc measures 50 degrees[/b][/color]. [br][b][color=#bf9000]What would the measure of the manila angle be? [br][/color][/b]
2.
[b][color=#ff00ff]Move ANY ONE (just ONE -- doesn't matter which) of the PINK POINTS[/color][/b] so the secant segment (for which this pink point is an endpoint) becomes TANGENT to the circle. [br][br]Answer question #1 again within THIS CONTEXT.
3.
Now [color=#ff00ff][b]m[/b][b]ove the pink points[/b][/color] so that BOTH secant segments become TANGENT SEGMENTS.[br]Suppose, in this case, the entire pink arc measures 200 degrees. [br][br][b][color=#1e84cc]What would the measure of the blue arc be? [/color][/b][br][color=#bf9000][b]What would the measure of the manila angle be? [/b][/color]
4.
Next, move the [b][color=#bf9000]MANILA POINT[/color][/b] (outside the circle) as close to the circle as possible so that the [b][color=#1e84cc]blue arc[/color][/b] almost disappears. (It won't disappear entirely). Keep the [b][color=#bf9000]MANILA POINT[/color][/b] on the circle. Now slowly re-slide the slider again. [br][br]What previously learned theorem do these transformations reveal? [br]
5.
Suppose the 2 secant segments (drawn from the [b][color=#bf9000]manila point[/color][/b] outside the circle) intersect the circle above so that the [b][color=#bf9000]manila angle measures 60 degrees[/color][/b] and the [b][color=#ff00ff]entire pink arc measures 200 degrees[/color][/b]. If this is the case, [color=#1e84cc][b]what would the measure of the entire blue arc be[/b][/color]?
– Tim Brzezinski

5.2.

–

5.3.

–

5.4.

–

5.5.

–

5.6.

–

5.7.

–

5.8.

–

5.9.

–

6.

Older (Earlier) Applets

  • 1. Inscribed Angle Theorem (Proof without Words)
  • 2. Inscribed Angle Theorem (Corollary 1) (Proof without Words)
  • 3. Inscribed Angle Theorem (Corollary 2) (Proof without Words)
  • 4. Inscribed Angle Theorem (Corollary 3) Cyclic Quadrilaterals
  • 5. Angle formed by a Secant & Tangent: What Do You See? (V1)
  • 6. Angle Formed by a Secant & Tangent: What Do You See? (V2)
  • 7. Angle Formed by 2 Tangents (V2): What Do You See?
  • 8. Cyclic Quadrilaterals
  • 9. Congruent Chords in the Same Circle
  • 10. Inscribed Angle Theorem (Intro): What Do You See Here?
  • 11. Inscribed Angle Action !!!
  • 12. Inscribed Angle Action! (2 Theorems)
  • 13. Angle Formed by 2 Chords (V1): What Do You See?
  • 14. Angle Formed by 2 Chords (V2): What Do You See?
  • 15. Chords Equidistant from Circle's Center
  • 16. Crossing Chords Property & Proof Start
  • 17. Crossing Chords Property & Proof Start (II)

6.1.

Inscribed Angle Theorem (Proof without Words)

Recall that the measure of an [color=#0a971e]arc[/color] of a circle is the same as the measure of its corresponding [color=#0a971e]central angle[/color]. (See applet.) [br][br][b]Definition[/b]: An [color=#b20ea8][b]INSCRIBED ANGLE[/b][/color] of a circle is an angle whose vertex lies on the circle and has each of its rays intersect the circle at one other point. (Click checkbox to show [color=#b20ea8]inscribed angle[/color].) [br][br]Notice how both the [color=#b20ea8]inscribed angle[/color] and [color=#0a971e]central angle[/color] both intercept the same [color=#0a971e]arc[/color]. [br][br]Click on the [color=#c51414]CHECK THIS OUT !!![/color] checkbox that appears afterwards. Be sure to move points [color=#1551b5]A[/color], [color=#1551b5]B[/color], and the [color=#b20ea8]pink vertex[/color] of the [color=#b20ea8]inscribed angle[/color] around. (You can also change the radius of the circle if you wish.)[br][br][b]In a circle, what is the relationship between the measure of an [color=#b20ea8]inscribed angle[/color] with respect to the measure of its [color=#0a971e]intercepted arc[/color]? [/b]
Inscribed Angle Theorem (Proof without Words)
Key directions and question are located above the applet.
– Tim Brzezinski

6.2.

–

6.3.

–

6.4.

–

6.5.

–

6.6.

–

6.7.

–

6.8.

–

6.9.

–

6.10.

–

6.11.

–

6.12.

–

6.13.

–

6.14.

–

6.15.

–

6.16.

–

6.17.

–

Information