Points of Concurrency
Points of Concurrency
Table of Contents
- Circumcenter
- Circumcenter & Circumcircle Action!
- Circumcenter Construction
- Incenter
- Incenter + Incircle Action (V2)!
- G.GCI.3 Constructing an Incenter: Ex. 11
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]
Incenter + Incircle Action (V2)!
In the app below, you can change the size of the triangle by moving any one (or more) of its vertices.[br][color=#ff00ff][b][br]You can also alter the size of the angle in the lower left corner by using the smaller slider. [br][/b][/color](You can also zoom in/out.)
1.
What vocabulary term would you use to describe the segments that have this triangle's vertices as its endpoints? Why?
2.
How do these 3 segments intersect? Describe.
3.
[b][color=#1e84cc]The point you see inside the triangle is called the INCENTER of this triangle.[/color][/b][br]Notice there are 3 equal distances in this triangle. How would you describe these equal distances in your own words? [br][br]That is, each of these distances is the distance from the _________ to ___________?
4.
Why do your observation(s) and response to (3) above hold true? What previously learned theorem justifies this?