Let us examine some important points related to a triangle. Use GeoGebra or any geometry software to follow the construction below.

1.) Construct triangle *TUV*.

2.) Construct the midpoint of each side.

3.) Construct the three altitudes. What do you observe? The intersection of the altitudes is called the *orthocente*r.

4.) Construct the three midpoints between the orthocenter and the three vertices. What do you observe about the nine points?

After finishing the steps above, your figure should at look like Figure 1. For the sake of discussion, we color the points. The red points are the midpoints of each side, the green points are the ‘foot’ of each altitude, and the cyan points are the midpoints of the orthocenter and and the vertices.

From the figure above, we observe two things: first, the altitudes seem to meet at a point; and second, it seems that the 9 points form a circle (can you verify this by construction?).

Using GeoGebra, we can verify if the nine points mentioned above lie on a circle. To do this, just select the *Circle through Three Points* tool, and then click any three of the nine points.

As we can see, the nine points are indeed on a circle . The circle is what we call the *nine-point circle*.

We find other interesting points on the circle.

5.) We construct a* circumcircle –*– a circle passing through vertices of the triangle and find its *circumcenter* (the center of the circumcircle). What can you say about the orthocenter, the center of the nine-point circle, and the circumcenter?

6.) We draw the *medians* — the segments that connect the three vertices to the midpoint of each side. What do you observe?

7.) We construct the *centroid* — the intersection of the medians.What do you observe?

*Euler line*in honor of Leonard Euler.

- The center of the nine-point circle is the midpoint of the orthocenter and the circumcenter.
- The radius of the circumcircle is twice that of the nine-point circle
- The incircle of a triangle is always inscribled to its nine-point circle.

Hi, thanks for making this interesting blog! Regarding the triangle, this is a such a good opportunity to also mention Napoleon’s theorem. Have a look at http://www.geogebra.org/en/upload/files/Norwegian/oisteing/trekantsenter.html for a GeoGegra file, where you can add regular triangles on the sides of your original triangle. (The five checkboxes at the bottom left)

Thanks Oistein. Thanks for sharing your applet. This is a good addendum to the post. I’ll make a link to your applet later.

Sir, may I ask about your expert opinion about a question in LET? I really hope you can help me with this question:

11) Performance in Mathematics Test

CLASS N MEAN STANDARD DEVIATION

A 40 57.30 8.92

B 35 57.08 10.17

C 40 58.12 11.85

D 39 58.25 10.28

Using the mean as basis for comparison, what can you say about the four classes in the above table?

a) Class D showed the best performance

b) Classes B, C and D showed about the same performance

c) Class D did better than Class C

d) Classes A and B have about the same performance.

Thank you po in advance. I really need this for a review session to my schoolmates this Sunday.

Hi Jaylord,

I’ll answer this on Saturday, we are quite busy now. Please remind me through my email: mathandmultimedia@gmail.com. 🙂

Guillermo

Thank you po for your response…I’ll look forward to it….:)

Dear Guillermo,

This is one of the best GeoGebra Lessons I have ever seen, you are a masterful teacher and you make learning fun and engaging and your questions to the students ( and/or the viewers, us 🙂 are challenging and thoughtful — you ask us to ‘muscle’ our brain and investigate new aspects of geometry with this fantastic internet tool (GeoGebra), to discover the magic and precision of geometry for ourselves.

Bravo! Thank you!

I’m adding your lesson link my High School Geometry Adventure wikispace…this is a gem.

Allen Berg

Dear Allen Berg,

You are most welcome and thank you very much for that inspiring message. I am glad that you have appreciated my work. 🙂

Guillermo

@Jaylord,

I am really sorry that I have not answered your question yesterday. I was very busy; and I forgot to check the recent comments.

Anyway, first, the question is a bit absurd because it seems that A, B, and C are

all correct. Obviously, the standard deviation in this question is irrelevant because the question says to ‘Use the mean as basis of comparison…’Here are my reasons why I think that the question was not well-written:

1.) It is obvious that option A is a correct answer since the class D has the highest mean.

2.) The scores of classes B, C, D, 57.3, 57.8, and 58.12, so basically, they showed about the same performance. So option B is also correct.

3.) Since class D performed best, it obviously performed better than class C, which makes option C correct.

4.) Option D is the only option that is ‘relatively’ incorrect since there is a 5-point difference between the means of class A and B.

I hope this helps.

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