Honors Geometry
Triangle Points of Concurrency & Euler’s Line
Name ____________________________
Due Date: Wed 1/16/13
Euler Line (Named after Swiss Mathematician, Leonhard Euler (1707-1783))
“The line segment that passes through a triangle’s orthocenter, centroid, and circumcenter - these three points are
collinear for any triangle.” (http://www.mathwords.com/e/euler_line.htm)
Goals of this Project:
A) Given three coordinate points that represent the vertices of a triangle find the coordinates of the
orthocenter, circumcenter and centroid of the triangle.
B) Using the solutions to part A, graph “The Euler’s Line” for your triangle.
C) Review the details of the geometry covered in Chapter 5.
D) Review important algebra concepts: Graphing, finding equations of lines given two points, finding
equations of perpendicular lines, midpoint formula and systems of equations
Assignment Details:
1. You will be randomly assigned three points in class to use as the three vertices of a triangle. Write them
here:
***COORDINATES OF VERTICES:
A(
,
),
B(
,
),
and C (
,
) ***
2. Materials needed
o At least 4 pieces of graph paper
o A ruler
o A protractor is helpful but not necessary
o A calculator
3.
The Centroid – in this part of the project, you will find the coordinates of the centroid of your triangle
both by sketching and estimating, and by using algebra to find the exact coordinates.
a. On a large x,y coordinate axes, plot your vertices and draw your triangle. Make sure to
label your vertices with A, B and C correctly, as well as with the coordinates.
b. Using your knowledge from Geometry class, carefully draw the three Medians of this
triangle. Note: You will need to calculate the coordinates of the midpoints of each side.
Label them as follows: Midpoint of AB is D, Midpoint of BC is E and Midpoint of AC is
F. USE A RULER TO DRAW THE MEDIANS.
c. Label the Centroid, and estimate it’s coordinates.
 involve a
d. You will now calculate the
coordinates of the Centroid algebraically. This will

significant amount of algebra. In order to get full credit for this part of the project, you
must show all of your work.
 You will need to find the equations of two of the medians. Please write the final equation of each
line on the correct median on the graph.
 Calculate the point of intersection of the two lines (using linear combinations or substitution –
systems of equations). Your solution to this system of equations will be the coordinates of the
Centroid. Compare your calculated Centroid with your estimated Centroid. They should be
reasonably close. Otherwise, you will need to double check your work and try again.
 Your calculations may involve a lot of fractions/decimals. Be very careful and round all
calculations to three decimal places to be as accurate as possible.
4.
The Circumcenter – in this part of the project, you will find the coordinates of the circumcenter of
your triangle both by sketching and estimating, and by using algebra to find the exact coordinates.
a. On a new large x,y coordinate axes, plot your vertices and draw your triangle. Make sure to label
your vertices with A, B and C correctly, as well as with the coordinates.
b. Using your knowledge from Geometry class, carefully draw the three Perpendicular Bisectors – one
on each side of this triangle. Note: You will need the midpoints again – so copy them from your
Centroid sketch. Label them the same way. To make sure that the line is perpendicular to the side
of the triangle, either use a protractor, or another source of a right angle (like the corner of the
ruler).
c. Label the Circumcenter, and estimate it’s coordinates.
d. You will now calculate the coordinates of the Circumcenter algebraically. This will involve a
significant amount of algebra. In order to get full credit for this part of the project, you must show
all of your work.
 You will need to find the equations of two of the perpendicular bisectors. Please write the final
equation of each line on the correct perpendicular bisector on the graph. To calculate the
equations of these lines, you will need to remember how to find the slope of a line if you know
the slope of a line that is perpendicular to it.
 Calculate the point of intersection of the two lines (using linear combinations or substitution –
systems of equations). Your solution to this system of equations will be the coordinates of the
Circumcenter. Compare your calculated Circumcenter with your estimated Circumcenter.
They should be reasonably close. Otherwise, you will need to double check your work and try
again.
 Your calculations may involve a lot of fractions/decimals. Be very careful and round all
calculations to three decimal places to be as accurate as possible.
5.
The Orthocenter – in this part of the project, you will find the coordinates of the orthocenter of your
triangle both by sketching and estimating, and by using algebra to find the exact coordinates.
a. On a new large x,y coordinate axes, plot your vertices and draw your triangle. Make sure to label
your vertices with A, B and C correctly, as well as with the coordinates.
b. Using your knowledge from Geometry class, carefully draw the three altitudes of the triangle – one
from each vertex. To make sure that the line is perpendicular to the side of the triangle, either use a
protractor, or another source of a right angle (like the corner of the ruler).
c. Label the Orthocenter, and estimate it’s coordinates.
d. You will now calculate the coordinates of the Orthocenter algebraically. This will involve a
significant amount of algebra. In order to get full credit for this part of the project, you must show
all of your work.
 You will need to find the equations of two of the altitudes. Please write the final equation of each
line on the correct altitude on the graph. To calculate the equations of these lines, you will need
to remember how to find the slope of a line if you know the slope of a line that is perpendicular
to it.
 Calculate the point of intersection of the two lines (using linear combinations or substitution –
systems of equations). Your solution to this system of equations will be the coordinates of the
Orthocenter. Compare your calculated Orthocenter with your estimated Orthocenter. They
should be reasonably close. Otherwise, you will need to double check your work and try again.
 Your calculations may involve a lot of fractions/decimals. Be very careful and round all
calculations to three decimal places to be as accurate as possible.
6. Once you are confident in your calculations of the coordinates of the circumcenter, orthocenter and centroid,
plot the three points on a new graph. If your calculations are correct, (according to Euler) these three
points should be collinear! Connect them to form the Euler’s Line for your triangle.
7. Reflection: Write a few sentences reflecting on the project – it’s educational value, your interest,
helpfulness, etc.
What you pass in:
1. Four accurately and neatly drawn graphs from steps 3-6 above.
2. All back up work in calculating equations of lines and using systems of equations to find points of
intersection.
3. Reflection about the project.
Honors Geometry Project – Euler’s Line
Please pass this in stapled to project
Name ______________________________
My three vertices are ___________________________________
Grading Checklist
This project will be worth 14 points (approximately equivalent to a quiz grade).
The Centroid (4 points)
o Graph is neatly drawn, labeled, medians are drawn correctly (2)
o Most lines/points are correctly drawn (1)
o Graph is incorrect (0)
------------------------------------------------------------------------------------------------o All work is included, neat, accurate and shows complete understanding (2)
o Most work is included, neat, accurate and shows complete understanding (1)
o Work does not show understanding, or is not included (0)
The Circumcenter (4 points)
o Graph is neatly drawn, labeled, perpendicular bisectors are drawn correctly (2)
o Most lines/points are correctly drawn (1)
o Graph is incorrect (0)
------------------------------------------------------------------------------------------------o All work is included, neat, accurate and shows complete understanding (2)
o Most work is included, neat, accurate and shows complete understanding (1)
o Work does not show understanding, or is not included (0)
The Orthocenter (4 points)
o Graph is neatly drawn, labeled, altitudes are drawn correctly (2)
o Most lines/points are correctly drawn (1)
o Graph is incorrect (0)
------------------------------------------------------------------------------------------------o All work is included, neat, accurate and shows complete understanding (2)
o Most work is included, neat, accurate and shows complete understanding (1)
o Work does not show understanding, or is not included (0)
Graph of Euler’s Line (1 point)
o Line is drawn accurately (1)
o Line is not drawn accurately (0)
Reflection (1 point)
GRADE: _______________