Euler Line Exploration SHS

Kaitlin, Kaiti Terpstra, Jan 10, 2023

Euler Line Exploration

Table of Contents

  1. Let's Review & Refresh!
    1. Constructing the Orthocenter
    2. Constructing the Circumcenter
    3. Constructing the Centroid
  2. Triangle Examples
    1. Triangle W
    2. Triangle X
    3. Triangle Y
  3. Leonhard Euler
    1. Euler Line
    2. DIY Euler line
  4. Summary & Extension
    1. Final Thoughts
    2. ... HOW?!

1.

Let's Review & Refresh!

These are a few concepts we will need to recap as we navigate through this activity. It is information you have already learned, but now we are going to apply it in some Geogebra constructions.

  • 1. Constructing the Orthocenter
  • 2. Constructing the Circumcenter
  • 3. Constructing the Centroid

1.1.

Constructing the Orthocenter

Recall...
The intersection of the three [b]altitudes[/b] of a triangle form the [b][color=#a64d79]orthocenter[/color][/b] of that triangle.
See if you can construct the orthocenter with what you already know. After about 5 minutes, move on.
Do you think you got it?
If you think you found it, explain how you did it and why you think it's the [b][color=#a64d79]orthocenter[/color][/b]. If you're not sure, just explain what you tried!
Compare the steps you described to the ones below.
1. Select one side of the triangle and create a line that is both perpendicular to that side and goes through the opposite vertex.[br][br]2. Repeat this step for each side so that you have the three [b]altitudes [/b]of the triangle.[br][br]3. Find the intersection of the three [b]altitudes[/b], and label it [b][color=#a64d79]O[/color][/b] for [b][color=#a64d79]orthocenter[/color][/b].
Was your construction method similar?
Try it again! This time with steps given.
– Kaitlin

1.2.

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1.3.

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2.

Triangle Examples

Using three triangles, W, X, and Y, we look at the triangle centers we worked on constructing in the previous chapter. Let's see what observations we make as we look through these examples.

  • 1. Triangle W
  • 2. Triangle X
  • 3. Triangle Y

2.1.

Triangle W

The colored lines represent different constructions of the three different triangle centers we are talking about. The intersection of the pink lines represent the orthocenter of triangle W.
The intersection of the green lines represent the circumcenter of triangle W.
The intersection of the blue lines represent the centroid of triangle W.
Here is all of them put together. Find each of these intersections, and label accordingly: orthocenter O, the circumcenter C, and the centroid N.
What observations do you have so far, if any, about the three centers of triangle W?
Do your best to use mathematically correct language as you share what you notice!
– Kaitlin

2.2.

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2.3.

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3.

Leonhard Euler

[i]It's pronounced "oil-er."[/i]

  • 1. Euler Line
  • 2. DIY Euler line

3.1.

Euler Line

Meet Leonhard Euler (pronounced "oil-er").
[size=100][size=150]Euler, on the hunt for something else, [i]accidentally[/i] discovered this fascinating fact about triangle centers. [/size][/size]
Looking back at the triangles from the activities, what information did the triangles and their centers have to tell us?
Well, the [color=#c27ba0]circumcenter[/color], [color=#6aa84f]orthocenter[/color], and [color=#3d85c6]centroid[/color] all fell on the same line!
Fun Fact:
The line that passes through the three distinguished points - [color=#a64d79]circumcenter[/color], [color=#6aa84f]orthocenter[/color], and [color=#3c78d8]centroid[/color] - is called the [b][color=#bf9000]Euler Line[/color][/b]. [br][br][size=85][i]There's a few other points on this line as well, but we will get to that in a moment.[/i][/size]
You mean to say that this is true for all triangles?
[size=150][b][br][br]I KNOW RIGHT![/b][/size]
Well... not all triangles.
The[b] [color=#bf9000]Euler line [/color][/b]exists for all triangles... except [u]one[/u] kind. As you watch the animation below, think back to the triangles from the previous activities. Was there ever a special case?
When [b]DON'T[/b] these three points create a line?
But wait, there's more!
The three points we talked about aren't the only ones on the Euler line. Aside from the orthocenter, circumcenter, and centroid, some other points that fall on this line are...[br][br]Center of the Nine Point Circle[br]Schiffler's Point [br]Exeter Point [br]Longchamps Point [br]Gossard Prospector[br]Incenter [color=#666666](only if the triangle is isosceles)[/color]
– Kaitlin

3.2.

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4.

Summary & Extension

  • 1. Final Thoughts
  • 2. ... HOW?!

4.1.

Final Thoughts

There was a lot of information thrown at you in these activities. Nice work!
[size=100][br][br]To help our brains organize what we learn, it's important to take a step back, summarize what you've learned, and allow yourself to think about any questions you may have. [/size]
First, let's summarize.
In 3-5 sentences, summarize what you've learned in these activities.
Questions? Ask away!
Come up with [b]at least one[/b] question. It can be one to help clear up any confusion, or it could be one to expand your new knowledge of the Euler line and/or triangle centers.
– Kaitlin

4.2.

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Information