Triangle Theorems

Tim Brzezinski, Sep 22, 2015

Table of Contents

  1. Theorems Involving Angles
    1. Triangle Angle Theorems
    2. Triangle Angle Theorems (V2)
    3. Triangle Angle Theorems (V3)
    4. Triangle Angle Sum Theorem (V4)
    5. Triangle Angle Sum Theorem
    6. Obvious Corollary
    7. Triangle Exterior Angle
    8. Angles of a Right Triangle
    9. Exterior Angles of a Triangle
  2. Triangle Theorems (General)
    1. Special Line through Triangle V1 (Theorem Discovery)
    2. Special Line through Triangle V2 (Theorem Discovery)
    3. Triangle Midsegment Action!
    4. A Special Triangle & Its Properties (I)
    5. Converse of IST (V1)
    6. Another Special Triangle and its Properties (II)
    7. Triangle Inequality: Discovery Lesson
    8. Triangle Side Possibilities? (V2)
    9. Triangle Inequality
    10. Triangle Inequality (II)
    11. Inequalities in 1 Triangle
    12. Another Special Right Triangle (Guided Discovery)
    13. Special Right Triangle (I)
    14. Acute, Right, or Obtuse?
    15. Acute, Right, Obtuse (III)
    16. Isosceles Triangle Medians
    17. Special Right Triangle (II)
    18. SAS: Dynamic Proof!
    19. SSS: Dynamic Proof!
    20. SSS: Dynamically Illustrated
    21. ASA Theorem?
    22. HL: Hypotenuse-Leg Action!
    23. Isosceles Triangle Theorem: Discovery Lab
    24. Geometric Mean Illustration
  3. Points of Concurrency
    1. Incenter Exploration (A)
    2. Incenter Exploration (B)
    3. Incenter & Incircle Action!
    4. Incenter + Incircle Action (V2)!
    5. Lines Containing Altitudes of a Triangle (V1)
    6. Orthocenter Exploration
    7. Circumcenter (& Questions)
    8. Circumcenter & Circumcircle Action!
    9. Triangle Medians: Quick Investigation
    10. Medians and Centroid Dance
    11. Medians Centroid Theorem (Proof without Words)
    12. Midpoint of HYP
    13. Points of Concurrency: Investigation
    14. Morley Action!
  4. Points of Concurrency - Extension Activities
    1. 3 Special Points!
    2. 9 Point Circle Action
    3. 9-Point Circle Action (Part 2)
    4. 9-Point Circle Action (Part 3A)
    5. 9-Point Circle Action (Part 3B)
    6. 9-Point Circle Action (Part 4)
    7. Euler Line Generator
  5. Other Triangle Theorems
    1. NCTM Calendar Problem (11-2-2016)
    2. Desargues Action!!!
  6. Older (Earlier) Applets
    1. Midsegment of a Triangle
    2. Isosceles Triangle (Properties)
    3. Euler Line (Informal Investigation)
    4. 9-Point Circle (Informal Investigation)

1.

Theorems Involving Angles

  • 1. Triangle Angle Theorems
  • 2. Triangle Angle Theorems (V2)
  • 3. Triangle Angle Theorems (V3)
  • 4. Triangle Angle Sum Theorem (V4)
  • 5. Triangle Angle Sum Theorem
  • 6. Obvious Corollary
  • 7. Triangle Exterior Angle
  • 8. Angles of a Right Triangle
  • 9. Exterior Angles of a Triangle

1.1.

Triangle Angle Theorems

Interact with the app below for a few minutes.  [br]Then, answer the questions that follow.  [br][br]Be sure to change the locations of this triangle's vertices each time [i]before[/i] you drag the slider!
What is the [b]sum of the measures of the interior angles of this triangle? [/b]
What is the [b]sum of the measures of the exterior angles [/b]of this triangle?
– Tim Brzezinski

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1.9.

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2.

Triangle Theorems (General)

  • 1. Special Line through Triangle V1 (Theorem Discovery)
  • 2. Special Line through Triangle V2 (Theorem Discovery)
  • 3. Triangle Midsegment Action!
  • 4. A Special Triangle & Its Properties (I)
  • 5. Converse of IST (V1)
  • 6. Another Special Triangle and its Properties (II)
  • 7. Triangle Inequality: Discovery Lesson
  • 8. Triangle Side Possibilities? (V2)
  • 9. Triangle Inequality
  • 10. Triangle Inequality (II)
  • 11. Inequalities in 1 Triangle
  • 12. Another Special Right Triangle (Guided Discovery)
  • 13. Special Right Triangle (I)
  • 14. Acute, Right, or Obtuse?
  • 15. Acute, Right, Obtuse (III)
  • 16. Isosceles Triangle Medians
  • 17. Special Right Triangle (II)
  • 18. SAS: Dynamic Proof!
  • 19. SSS: Dynamic Proof!
  • 20. SSS: Dynamically Illustrated
  • 21. ASA Theorem?
  • 22. HL: Hypotenuse-Leg Action!
  • 23. Isosceles Triangle Theorem: Discovery Lab
  • 24. Geometric Mean Illustration

2.1.

Special Line through Triangle V1 (Theorem Discovery)

Interact with the following applet for a few minutes. [br][br]As you do, be sure to change the locations of the triangle's vertices each time before (and even after) sliding the slider. [br][br]Then, answer the questions that follow.
1.
How would you describe the FIRST small white point that appeared as you animated the diagram?
2.
What can you conclude about the gray line and the gray side of the triangle? Why can you conclude this?
3.
What can you conclude about a line that passes through the midpoint of one side of a triangle and is parallel to another side of the triangle?
4.
Suppose you know that a line that passes through the midpoint of one side of a triangle and is parallel to another side of the triangle. Write a formal argument that proves this line [b]MUST[/b] pass through the [color=#9900ff][b]midpoint of the 3rd side[/b][/color] of the triangle.
– Tim Brzezinski

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3.

Points of Concurrency

  • 1. Incenter Exploration (A)
  • 2. Incenter Exploration (B)
  • 3. Incenter & Incircle Action!
  • 4. Incenter + Incircle Action (V2)!
  • 5. Lines Containing Altitudes of a Triangle (V1)
  • 6. Orthocenter Exploration
  • 7. Circumcenter (& Questions)
  • 8. Circumcenter & Circumcircle Action!
  • 9. Triangle Medians: Quick Investigation
  • 10. Medians and Centroid Dance
  • 11. Medians Centroid Theorem (Proof without Words)
  • 12. Midpoint of HYP
  • 13. Points of Concurrency: Investigation
  • 14. Morley Action!

3.1.

Incenter Exploration (A)

[color=#000000]Recall that 3 or more lines are said to be concurrent if and only if they intersect at exactly one point. [br][br]The angle bisectors of a triangle's 3 interior angles are all concurrent. [br]Their point of concurrency is called the I[b]NCENTER[/b] of the triangle. [br][br]In the applet below, [b]point I [/b]is the triangle's [b]INCENTER[/b]. [br]Use the tools of GeoGebra in the applet below to complete the activity below the applet. [br][i]Be sure to answer each question fully as you proceed. [/i] [/color]
[color=#000000][b]Directions: [/b][br][br][/color][left][color=#000000]1) In the applet above [/color][color=#38761d]construct a line passing through I and is perpendicular to [i]AB[/i][/color][color=#000000]. [br]2) Use the [/color][b][color=#000000]Intersect[/color][/b][color=#000000] tool to plot and label a point [/color][i]G[/i][color=#000000] where [/color][color=#38761d]the line you constructed in (1)[/color][color=#000000] intersects [/color][i][/i][color=#000000][i]AB[/i].[/color][color=#000000][br]3) [/color][color=#38761d]Construct a line that passes through I and is perpendicular to [i]BC[/i].[/color][color=#000000] [br][/color][color=#000000]4) Plot and label a point [i]H[/i] where [/color][color=#38761d]the line you constructed in (3)[/color][color=#000000] intersects [i]BC[/i].[/color][br][color=#000000]5) [/color][color=#38761d]Construct a line that passes through I and is perpendicular to [i]A[/i][i]C[/i]. [/color][color=#000000][br][/color][color=#000000]6) Plot and label a point [/color][i]J[/i][color=#000000] where [/color][color=#38761d]the line you constructed in (5)[/color][color=#000000] intersects [/color][i][color=#000000]AC[/color][/i][color=#000000]. [br]7) Now, use the [b]Distance[/b] tool to measure and display the lengths [i]IG[/i], [i]IH[/i], and [i]IJ[/i]. What do you notice?[br][br][br]8) Experiment a bit by moving any one (or more) of the triangle's vertices around[br] Does your initial observation in (7) still hold true? [br] Why is this? (If you need a hint, refer back to the worksheet found [url=https://tube.geogebra.org/m/tU3ZqhjN]here[/url]. [/color][/left][color=#000000][br]9) Construct a circle centered at I that passes through [i]G[/i]. What else do you notice? [br] Experiment by moving any one (or more) of the triangle's vertices around. [br] This circle is said to be the triangle's [i]incircle[/i], or [i]inscribed circle[/i]. [br] It is the largest possible circle one can draw [i]inside[/i] this triangle. [br] Why, according to your results from (7) is this possible? [br][br]10) Do the angle bisectors of a triangle's interior angles also bisect the sides opposite theses angles? [br] Use the [b]Distance[/b] tool to help you answer this question. [/color][br][br][color=#000000]11) Is it ever possible for a triangle's [b]INCENTER[/b] to lie OUTSIDE the triangle?[br] If so, under what condition(s) will this occur? [br][br]12) Is it ever possible for a triangle's [b]INCENTER[/b] to lie ON the triangle itself?[br] If so, under what condition(s) will this occur? [/color]
– Tim Brzezinski

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4.

Points of Concurrency - Extension Activities

  • 1. 3 Special Points!
  • 2. 9 Point Circle Action
  • 3. 9-Point Circle Action (Part 2)
  • 4. 9-Point Circle Action (Part 3A)
  • 5. 9-Point Circle Action (Part 3B)
  • 6. 9-Point Circle Action (Part 4)
  • 7. Euler Line Generator

4.1.

3 Special Points!

[color=#000000]Recall the following: [/color][br][br][color=#ff7700]1) The lines that contain a triangle's 3 altitudes are concurrent (intersect at exactly one point.) [br] This point of concurrency is called the orthocenter of the triangle. [/color][br][br][color=#cc0000]2) A triangle's 3 perpendicular bisectors are concurrent at a point called the circumcenter of the triangle.[/color][br][br][color=#38761d]3) A triangle's 3 medians are concurrent at a point called the centroid of the triangle. [br][br][/color][color=#000000]Interact with the applet below for a few minutes. Then answer the discussion questions that follow. [/color]
[color=#000000][b]Questions: [/b][br][br]1) What conclusion can you make about the positioning of a triangle's[/color] [color=#ff7700]orthocenter,[/color] [color=#cc0000]circumcenter,[/color] [color=#000000]and[/color] [br] [color=#38761d]centroid[/color][color=#000000]? Explain how you can use the toolbar to illustrate this. [/color][br][br][color=#000000]2) How does the sliding the slider also informally show that your response to (1) is true? [/color][br][br][color=#000000]3) Let's denote the [/color][color=#ff7700]orthocenter as [/color][i][color=#ff7700]O[/color], [/i][color=#000000]the [/color][color=#cc0000]circumcenter as [i]C[/i][/color], [color=#000000]and the[/color] [color=#38761d]centroid as [i]G[/i][/color][color=#000000].[/color] [br] [color=#000000]What is the exact value of the ratio [i]CG/CO[/i]? What is the exact value of the ratio [i]CG[/i]/[i]GO[/i]?[/color] [br][br][color=#000000]4) Prove your assertion for (1) true using a coordinate geometry format. [br] For simplicity's sake, position the triangle so its vertices have coordinates (0,0), (6a, 0), and (6b, 6c). [br][br]5) Prove your responses to (3) are true using the same coordinate geometry setup you used in (4) above. [br][br]6) Research information about the Euler Line of a triangle. [br] How does the Euler Line relate to the context of the above applet? [/color]
– Tim Brzezinski

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5.

Other Triangle Theorems

  • 1. NCTM Calendar Problem (11-2-2016)
  • 2. Desargues Action!!!

5.1.

NCTM Calendar Problem (11-2-2016)

Creation of this applet was inspired by the the calendar problem (11/2/2016) that appeared in [url=http://www.nctm.org/]NCTM[/url]'s November 2016 issue of the [url=http://www.nctm.org/publications/mathematics-teacher/][i]Mathematics Teacher[/i] magazine[/url]. [br][br]"In right triangle [i]ABC[/i],[color=#0000ff][b] [i]CD[/i] is the bisector[/b][/color] of [color=#666666][b]right angle [i]ACB[/i][/b][/color], [color=#9900ff][b][i]CM[/i] is the median to the hypotenuse[/b][/color], and [color=#980000][b][i]CP [/i]is the altitude to the hypotenuse[/b][/color]. Prove [i][color=#0000ff][b]CD[/b][/color][/i] [color=#ff00ff][b]bisects[/b][/color] angle [i]PCM[/i]. " [br][br]This applet informally illustrates what this calendar problem is asking you to prove. [br][br]Can you formally prove what this applet informally illustrates?
– Tim Brzezinski

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6.

Older (Earlier) Applets

  • 1. Midsegment of a Triangle
  • 2. Isosceles Triangle (Properties)
  • 3. Euler Line (Informal Investigation)
  • 4. 9-Point Circle (Informal Investigation)

6.1.

Midsegment of a Triangle

Definition: A [b]midsegment of a triangle[/b] is a segment that connects the midpoints of any 2 sides of that triangle. [br]Question: How many midsegments does a triangle have? [br][br]Let's proceed:[br][br]In the applet below, points [color=#1551b5]D[/color] and [color=#c51414]E[/color] are midpoints of 2 sides of triangle ABC. One [color=#0a971e]midsegment[/color] of Triangle ABC is shown in [color=#0a971e]green[/color]. [br]Move the vertices A, B, and C of Triangle ABC around. As you do, observe the two comments off to the right side. [br]Then, answer the questions below the applet.
Questions: [br][br]1) What do you notice about the slopes of segments [color=#0a971e]DE[/color] and AB? What does this imply about these 2 segments? [br]2) What does the ratio of [color=#0a971e]DE[/color] to AB tell us about the [color=#0a971e]midsegment [/color]of any triangle? [br][br]3) If we refer to the black side of the triangle as the triangle's "3rd side", complete the following statement. Be sure to use the phrase "3rd side" in each blank below. [br][br] [b]The [color=#0a971e]MIDSEGMENT of a triangle[/color] is ALWAYS [br][br] i) ________________________________________________________________________, and[br][br] ii) ________________________________________________________________________. [/b]
– Tim Brzezinski

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Information