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Triangles: Points of Concurrency

Tim Brzezinski, Maria Green, Nov 5, 2017

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

Prerequisites
1. Triangle Altitude Illustrator and Definition Writing Prompt
2. Perpendicular Bisector Definition
3. Perpendicular Bisector Definition
4. Angle Bisector Definition (I)
5. Triangle Median Illustrator & Definition Writing Prompt
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 3B)
9. 9-Point Circle Action (Part 4)
10. Vivani Action!
11. Napoleon Motion!
12. GoGeometry Action 17!
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)
6. Triangle Medians: Quick Investigation

Information

Prerequisites

1. Triangle Altitude Illustrator and Definition Writing Prompt
2. Perpendicular Bisector Definition
3. Perpendicular Bisector Definition
4. Angle Bisector Definition (I)
5. Triangle Median Illustrator & Definition Writing Prompt

Triangle Altitude Illustrator and Definition Writing Prompt

Interact with the applet for a few minutes. The purple segment that will appear is said to be an ALTITUDE OF A TRIANGLE. Be sure to move the blue vertex of the triangle around a bit as well. Answer the questions that appear below the applet.

1.

Is it ever possible for a triangle's altitude to lie inside the triangle?

2.

Is it ever possible for a triangle's altitude to lie ON the triangle itself? That is, can an altitude of a triangle ever be the same as ONE SIDE of the triangle? If so, in what kind of a triangle will this occur?

3.

Is it ever possible for a triangle's altitude to lie entirely OUTSIDE the triangle?

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): An altitude of a triangle is...

1.2.

1.3.

1.4.

1.5.

2.

Dynamic Illustrations (no words or data)!

2.1.

Circumcenter & Circumcircle Action!

Interact with this applet for a few minutes, then answer the questions that follow. Be sure to change the locations of the triangle's VERTICES both BEFORE and AFTER sliding the slider! In addition, note the pink slider controls the measure of the interior angle with pink vertex (lower left).

1.

What can you conclude about the 3 smaller blue points? What are they? How do you know this?

2.

What vocabulary term best describes each brown line? Why is this?

3.

Describe the intersection of these 3 brown lines. How do they intersect?

The ORANGE POINTis called the CIRCUMCENTER of the triangle. Also, note that the pink slider controls the measure of the interior angle with pink vertex (lower left).

6.

Is it ever possible for the circumcenter to lie outside the triangle? If so, how would you classify such a triangle by its angles?

7.

Is it ever possible for the circumcenter to lie on the triangle itself? If so, how would you classify such a triangle by its angles? And if so, where exactly on the triangle is the circumcenter found?

8.

Is it ever possible for the circumcenter to lie inside the triangle? If so, how would you classify such a triangle by its angles?

9.

What is so special about the purple circle with respect to the triangle's vertices?

10.

What previously learned theorem easily implies that the distance from the circumcenter to any vertexis equal to the distance from the circumcenter to any other vertex?

2.2.

2.3.

2.4.

2.5.

2.6.

2.7.

2.8.

2.9.

2.10.

2.11.

2.12.

3.

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)
6. Triangle Medians: Quick Investigation

3.1.

Orthocenter Exploration

Recall that 3 or more lines are said to be concurrent if and only if they intersect at exactly 1 point.

Here, the triangle's 3 vertices are MOVEABLE. Slide the bottom slider really slowly and carefully observe what is taking place.

The point O you see is said to be the orthocenter of the triangle. What do you notice? What do you wonder? Describe!

Is it possible for the orthocenter of a triangle to lie INSIDE THE TRIANGLE? If so, under what condition(s) do/does this occur?

Is it possible for the orthocenter of a triangle to lie ON THE TRIANGLE ITSELF? If so, under what condition(s) do/does this occur?

Is it possible for the orthocenter of a triangle to lie OUTSIDE THE TRIANGLE? If so, under what condition(s) do/does this occur?

Without Googling, how would you define the term ORTHOCENTER OF A TRIANGLE? Describe.

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