Geometry
Concurrency Points Investigation
Name__________________
Oct. 12/13, 2011
Directions: In order to get the most from the investigation, please complete the steps in order. If you are
experienced with Geometer’s Sketchpad, you can skip reading the italicized directions within the
parentheses.
Working with Triangles & Perpendicular Bisectors
Open Sketchpad and go to File—Save As and save the document on your Desktop as Concurrency
Points. Periodically save this document as you work your way through the following investigations.
1.
Construct any triangle (use the segment maker in the side bar or make three points and construct the
sides by drawing three points, highlighting all three, and then go to the Construct menu and look for
“segments”).
2. Construct the perpendicular bisectors of the sides of your triangle (highlight a side and construct
“midpoint”, then highlight the side and its midpoint, and construct “perpendicular”).
3. Are the three perpendicular bisectors concurrent (i.e. do they all intersect at a single point)?
To better distinguish the perpendicular bisectors from your triangle segments, highlight all three
perpendicular lines, go to Display—Line Style—Dashed.
4. Show that the point of concurrence is equidistant from the vertices of the triangle (click on the circle
marker on the side bar and construct a circle with its center at this concurrency point so that its
radius connects to one of the vertices of the triangle).
5. Drag one of the vertices of your triangle around to make it take on many shapes and sizes. Take note
of any observations you would consider noteworthy.
6. The point of concurrence of the perpendicular bisectors is called the circumcenter. Why do you think
it has this name?
7. Label the circumcenter as ‘Ci’ (highlight two of the perpendicular bisectors and then construct
“intersection” or simply place a point from the sidebar at the intersection, then with only the
intersection point highlighted, go to display and choose “label point”).
Geometry
Concurrency Points Investigation
Name__________________
Oct. 12/13, 2011
Before moving on to the next investigation, go to File—Document Options—Add Page—Blank Page—
OK. This will enable you to go back and play with your prior investigation at any point. You should
notice at the bottom left-hand corner of your screen you now have two pages.
Working with Triangles & Medians
8. Construct any triangle.
9. Construct the three medians for your triangle. Recall what it means for a line to be a median of a
triangle.
10. Are the medians concurrent?
To better distinguish these medians from your triangle segments, highlight all three medians, go to
Display—Line Style—Dashed.
11. Drag one of the vertices of your triangle around to make it take on many shapes and sizes. Take note
of any observations you would consider noteworthy.
12. The point of intersection of the medians is called the centroid. This point has a very interesting
property! Feel free to investigate on your own or ask a physics teacher…
13. Label the centroid ‘Ce’.
Add a Blank Page.
Working with Triangles & Angle Bisectors
14. Construct any triangle.
15. Construct the angle bisectors of the triangle’s three angles (highlight a point on a side of the angle,
then the vertex, and then a point on the other side. Now go to construct “angle bisector”).
16. Are the angle bisectors concurrent?
To better distinguish these angle bisectors from your triangle segments, highlight all three angle
bisectors, go to Display—Line Style—Dashed.
Geometry
Concurrency Points Investigation
Name__________________
Oct. 12/13, 2011
17. Drag one of the vertices of your triangle around to make it take on many shapes and sizes. Take note
of any observations you would consider noteworthy.
18. The point of concurrence of the angle bisectors is called the incenter.
19. Label the incenter as ‘In’.
20. Construct a perpendicular from the incenter to one of the sides of the triangle. Construct the point of
intersection between this perpendicular and the side it is perpendicular to. Now construct a circle by
center and point with the incenter as the center and the intersection of the perpendicular and the side
as the point.
21. Why do you think this point of concurrence is called the incenter?
Add a Blank Page.
Working with Triangles & Altitudes
22. Construct any triangle.
23. Construct the three altitudes of the triangle. Label their point of concurrence as ‘Or’ for orthocenter.
Why might this point be called the orthocenter?
To better distinguish these altitudes from your triangle segments, highlight all three altitudes, go to
Display—Line Style—Dashed.
Add a Blank Page.
Geometry
Name__________________
Concurrency Points Investigation
Oct. 12/13, 2011
Summary of the Points of Concurrence (related to a triangle)
Do some remarkable connections exist between these points of concurrence? Let’s find out…
24. Construct a triangle. Construct all points of concurrency on this one triangle. This will take a little
time, but you now have the skills to do this!
Advice: After constructing a point of concurrence and labeling it appropriately (don’t forget to
do this!), hide the lines used to create the point of concurrence. For example, to hide the
perpendicular bisectors, highlight only the three perpendicular bisectors and then go to the
display menu and choose “hide perpendiculars”). This will allow you to only track the
circumcenter, ‘Ci’ and then move on to your next construction.
25. Which of the points of concurrence (circumcenter, centroid, incenter, & orthocenter) are always in
the interior of the triangle?
26. Look carefully at a variety of triangles where the four concurrence points are all different. Three of
the four always turn out to be collinear. Which ones?
Note: The line containing these points is called the Euler line (pronounced like “oiler”) after the 18th
century German mathematician who first observed it.
27. As you look at various triangles, do the points on the Euler line always stay in the same order?
28. Measure all the distances between concurrency points along the Euler line (there are three distances in
all). These distances always have specific numerical ratios.
Find the ratios and express the relationship by filling in the blanks (you can find ratios after
measuring lengths by going to “Number” and then “Calculate”. You can highlight the top of the
ratio, then the division symbol, and then highlight the bottom of the ratio):
29. Which of the concurrence point(s) divides the segments used to create it into a 2-to-1 ratio?
“The ________________ divides the Euler line into two parts so that
the smaller part is _______________ the larger part.”
30. Make observations about any of the points of concurrence for right triangles, equilateral triangles,
isosceles triangles, obtuse triangles, acute triangles, etc.?
Geometry
Name__________________
Concurrency Points Investigation
Oct. 12/13, 2011
Point of Concurrence Summary
Name of Center
Centroid
Incenter
Circumcenter
Orthocenter
How it is formed
Interesting Properties