Triangles: Points of Concurrency
Circumcenter, Orthocenter, Centroid, Incenter, Perpendicular Bisectors, Altitudes, Medians, Angle Bisectors
Table of Contents
- Prerequisites
- Triangle Altitude Illustrator and Definition Writing Prompt
- Perpendicular Bisector Definition
- Perpendicular Bisector Definition
- Angle Bisector Definition (I)
- Triangle Median Illustrator & Definition Writing Prompt
- Dynamic Illustrations (no words or data)!
- Circumcenter & Circumcircle Action!
- Incenter & Incircle Action!
- Medians and Centroid Dance
- Incenter + Incircle Action (V2)!
- Medians & Equal Areas!
- Applets with Numerical Data
- Lines Containing Altitudes of a Triangle (V1)
- Orthocenter Exploration
- Circumcenter (& Questions)
- Medians Centroid Theorem (Proof without Words)
- Incenter Exploration (B)
- Triangle Medians: Quick Investigation
Triangle Altitude Illustrator and Definition Writing Prompt
Interact with the applet for a few minutes. [br][br]The [color=#9900ff][b]purple segment[/b][/color] that will appear is said to be an [b][color=#9900ff]ALTITUDE OF A TRIANGLE.[/color][/b] [br]Be sure to move the [b][color=#1e84cc]blue vertex[/color][/b] of the triangle around a bit as well. [br][br]Answer the questions that appear below the applet.
1.
Is it ever possible for a triangle's [b]altitude[/b] to lie [b][color=#9900ff]inside the triangle[/color][/b]?
2.
Is it ever possible for a triangle's [b][color=#9900ff]altitude[/color][/b] to lie ON the triangle itself? [br]That is, can an [b][color=#9900ff]altitude[/color][/b] of a triangle [b]ever be the same as ONE SIDE of the triangle[/b]? [br][br]If so, in what kind of a triangle will this occur?
3.
Is it ever possible for a triangle's [b][color=#9900ff]altitude[/color][/b] to lie[b] entirely OUTSIDE the triangle[/b]?
4.
Given your responses to these question and what you've observed, complete the following sentence definition (without looking it up on another tab in your browser): [br][br][i]An [b][color=#9900ff]altitude[/color][/b] of a triangle is...[/i]
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]