Incenter Investigation

Euler Line Investigation

Background

In this section of the exploration you will be asked to discover the amazing property that certain points of concurrency are collinear, i.e. lie on the same line, independent of the configuration of the triangle. Your investigation will result in a discovery, known long ago before people could use computers. This discovery was named after Euler, a great mathematician and scientist, whose profound work and genius have greatly influenced the development of our understanding of the world. Thus using the power of modern technology, you will be able to understand properties that have puzzled some of the greatest minds of the past. So, try hard and patiently and you will be rewarded with intriguing discoveries. As you work through the material, you may be asked to write down the answers to the questions in the investigation or to save the information and your sketch on a disk.

We start our journey with our old friend -- the triangle. There is nothing simpler than a triangle, just three points and segments that connect them, but what a world of unexpected relations! You have already been introduced to some of them -- the four points of concurrency. Could you name them? If you have doubts about the names of the points and their definitions, check here.

Investigation

1. Click once here to open a sketch with our old friend -- triangle ABC.

2. Construct the orthocenter of triangle ABC. After the construction, hide the altitudes but leave the orthocenter.If you are in doubt about the construction click here to review the definitions.

3. Construct the circumcenter of triangle ABC. After the construction hide all perpendicular bisectors, but leave the circumcenter. If you are in doubt about the construction click here to review the definitions.

4. Construct the incenter of triangle ABC. After the construction hide all bisectors but leave the incenter. If you are in doubt about the construction click here to review the definitions.

5. Construct the centroid of triangle ABC. After the construction hide all lines but leave the centroid. If you are in doubt about the construction click here to review the definitions.

6. At this point your picture should look approximately like this:

Until now you had to be very patient. The real fun is just beginning.

Try to see if any three of the points lie on the same line. A strategy for doing this is to construct a line through any two of the points and to check if a third point is lying on this line when you change the triangle. If your first line does not work, try other lines.
Which three points lie on the line? Make a conjecture.
Check your conjecture again by moving the triangle. If it holds you have discovered the Euler line .

We are not done yet. We still have properties to discover. Follow the directions ahead.

Connect the circumcenter to the centroid with a segment. Measure its length.
Connect the centroid to the orthocenter with a segment. Measure its length.
Compare the lengths by finding their ratio. Change the triangle. What happens to the ratio? Make a conjecture!
Click here to check your answer.

Now you can relax. You have discovered some of the properties of the Euler's line. You will discover more properties when you investigate the nine point circle.

Do you know that mathematician do not stop here? This is just the beginning of another world, where mathematicians enter with conjectures and come out with a proof! Conjectures are not always true! You have to prove them with logic and serious argument. To do so you need to know the rules, be very persistent and study a lot more mathematics... This is what mathematicians do!

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