Triangles: Points of Concurrency

Tim Brzezinski, Mark Corpuz, Dec 2, 2016

Circumcenter, Orthocenter, Centroid, Incenter, Perpendicular Bisectors, Altitudes, Medians, Angle Bisectors, Euler Line, 9-Point Circle

Table of Contents

  1. Dynamic Illustrations (no words or data)!
    1. Circumcenter & Circumcircle Action!
    2. Incenter & Incircle Action!
    3. Medians and Centroid Dance
    4. Medians & Equal Areas!
    5. 9 Point Circle Action
    6. 9-Point Circle Action (Part 2)
    7. 9-Point Circle Action (Part 3A)
    8. 9-Point Circle Action (Part 4)
    9. Vivani Action!
    10. Napoleon Motion!
    11. 9-Point Circle Action (Part 3B)
  2. Applets with Numerical Data
    1. Orthocenter Exploration
    2. Circumcenter (& Questions)
    3. Medians Centroid Theorem (Proof without Words)
    4. Incenter Exploration (B)
    5. 9-Point Circle (Informal Investigation)

1.

Dynamic Illustrations (no words or data)!

  • 1. Circumcenter & Circumcircle Action!
  • 2. Incenter & Incircle Action!
  • 3. Medians and Centroid Dance
  • 4. Medians & Equal Areas!
  • 5. 9 Point Circle Action
  • 6. 9-Point Circle Action (Part 2)
  • 7. 9-Point Circle Action (Part 3A)
  • 8. 9-Point Circle Action (Part 4)
  • 9. Vivani Action!
  • 10. Napoleon Motion!
  • 11. 9-Point Circle Action (Part 3B)

1.1.

Circumcenter & Circumcircle Action!

[color=#000000]Interact with this applet for a few minutes, then answer the questions that follow. [br][br]Be sure to change the locations of the triangle's [/color][b]VERTICES[/b] both [b]BEFORE[/b] and [b]AFTER[/b] sliding the slider![br]In addition, note the [b][color=#ff00ff]pink slider[/color][/b] controls the measure of the interior angle with [b]pink vertex (lower left)[/b].
1.
What can you conclude about the [b][color=#1e84cc]3 smaller blue points[/color][/b]? What are they? How do you know this?
2.
[color=#000000]What vocabulary term best describes each [/color][color=#980000][b]brown line[/b][/color][color=#000000]? Why is this? [/color]
3.
[color=#000000]Describe [/color][b][color=#ff7700]the intersection[/color][/b][color=#000000] of these [/color][color=#980000][b]3 brown lines[/b][/color][color=#000000]. [/color][b][color=#ff7700]How do they intersect?[/color][/b]
[color=#ff7700][b]The ORANGE POINT[/b][/color]is called the [b][color=#ff7700]CIRCUMCENTER[/color][/b][color=#000000] of the triangle. [br][/color]Also, note that the [b][color=#ff00ff]pink slider[/color][/b] controls the [b][color=#ff00ff]measure of the interior angle with pink vertex[/color][/b] (lower left).
6.
[color=#000000]Is it ever possible for the [/color][b][color=#ff7700]circumcenter [/color][/b][color=#000000]to lie [i]outside the triangle[/i]?[br]If so, how would you classify such a triangle by its angles? [/color]
7.
[color=#000000]Is it ever possible for the [/color][color=#ff7700][b]circumcenter[/b] [/color][color=#000000]to lie [i]on the triangle itself[/i]?[br]If so, how would you classify such a triangle by its angles? [br]And if so, [i]where exactly on the triangle[/i] is the [/color][b][color=#ff7700]circumcenter[/color][/b][color=#000000] found? [/color]
8.
[color=#000000]Is it ever possible for the [/color][b][color=#ff7700]circumcenter[/color][/b][color=#000000] to lie [i]inside the triangle[/i]?[br]If so, how would you classify such a triangle by its angles? [/color]
9.
[color=#000000]What is so special about the [/color][b][color=#9900ff]purple circle [/color][/b][color=#000000]with respect to the triangle's vertices[/color][color=#000000]? [/color]
10.
[color=#000000]What [/color][color=#ff00ff][b]previously learned theorem[/b][/color][color=#000000] easily implies that the distance from the [/color][b][color=#ff7700]circumcenter[/color][/b][color=#000000] to any [/color]vertex[color=#000000]is equal to the distance from the [/color][b][color=#ff7700]circumcenter[/color][/b][color=#000000] to any other [/color]vertex[color=#000000]? [/color]
– Tim Brzezinski

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1.9.

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1.10.

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1.11.

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2.

Applets with Numerical Data

  • 1. Orthocenter Exploration
  • 2. Circumcenter (& Questions)
  • 3. Medians Centroid Theorem (Proof without Words)
  • 4. Incenter Exploration (B)
  • 5. 9-Point Circle (Informal Investigation)

2.1.

Orthocenter Exploration

[color=#000000]Recall that 3 or more lines are said to be concurrent if and only if they intersect at exactly 1 point. [/color]
Here, the triangle's 3 vertices are MOVEABLE. Slide the bottom slider really slowly and carefully observe what is taking place.
The point [i][b]O[/b][/i] you see is said to be the [b]orthocenter[/b] of the triangle. What do you notice? What do you wonder? Describe!
Is it possible for the [b]orthocenter[/b] of a triangle to lie [b]INSIDE THE TRIANGLE?[/b] If so, under what condition(s) do/does this occur?
Is it possible for the [b]orthocenter[/b] of a triangle to lie [b]ON THE TRIANGLE ITSELF? [/b] If so, under what condition(s) do/does this occur?
Is it possible for the [b]orthocenter[/b] of a triangle to lie [b]OUTSIDE THE TRIANGLE? [/b] If so, under what condition(s) do/does this occur?
Without Googling, how would you define the term [b]ORTHOCENTER OF A TRIANGLE? [/b]Describe.
– Tim Brzezinski

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