Geometry: 2nd Semester (or 4th Quarter) Project Options
You must choose one of the following three options for your geometry project this semester. Regardless of
which option you choose, the project will be worth 100 points and make up 10% of your semester grade so
do a nice job!!
Option 1: The Nine-Point Circle
The Nine-Point Circle has been attributed to Leonhard Euler in the 1700s, but others attribute it to Karl
Feuerbach, who published a proof about the circle in 1822. Your task is to do some research on the circle,
Euler, and Feuerbach and then construct it for three triangles.
1. Carefully follow the steps below to construct the Nine-Point Circle on graph paper. Hint: Erase
unnecessary lines as you go to prevent confusion. (Refer to p. 263 and chapter 5, Properties of Triangles, in
your textbook)
a. Draw a large, acute, scalene triangle and label the vertices X, Y, and Z.
b. Construct the perpendicular bisectors of the sides of the triangle. Label the midpoints of the sides
A, B, and D. Label the point where the perpendicular bisectors intersect as C, the circumcenter.
c. Using the midpoints A, B, and D, draw the medians of the triangle. Label the point where the
medians intersect as E, the centroid.
d. Construct an altitude from each vertex to the opposite side. Label the point where the altitudes
intersect the sides as H, J, and K. Label the point where the medians meet as O, the orthocenter.
e. Draw the segments OX, OY, and OZ. Find the midpoint of each segment and label the points L, M,
and N.
f. Construct the angle bisector of each angle of ∆ XYZ . Label the point of intersection of the angle
bisectors as I, the incenter.
g. Draw OC. Bisect the segment and label the midpoint T.
h. Draw a circle with center T and radius AT.
i. If you have done the constructions carefully, the points A, B, D, H, J, K, L, M, and N should lie on
(or very close to) a circle.
2. Construct two more Nine-Point Circles, one for a right triangle, and one for an obtuse triangle.
3. Write several sentences comparing the Nine-Point Circle for an acute, a right, and an obtuse triangle.
Option 2: Sierpinski Triangles
The Sierpinski Triangle is a fractal, a figure created by repeating the same process over and over again. Do
research on the Sierpinski Triangle and fractals. How are fractals developed? What are their uses or
occurrences in the natural world? Then construct stage 0 to stage 4 Sierpinski Triangles by using the
following process. (Refer to the bottom of p. 291 and p. 590-591 of your textbook for more information)
a. Draw an equilateral triangle with each side 16 units long.
b. Connect the midpoints of each side to form another triangle. Shade the center triangle.
c. Repeat the process using the three non-shaded triangles. Connect the midpoints of each side to
form other triangles.
Option 3: Centroids, Orthocenters, Circumcenters, and the Euler Line.
The ancient Greeks knew about the classical triangle centers, but missed a crucial relationship between
them. The relationship was discovered by Leonhard Euler in the 18th century. His discovery revived interest
in Euclidean geometry. Research the life of Euler and triangle centers. Then construct the following figures.
(Refer to p. 263 and chapter 5, Properties of Triangles, in your textbook)
a. Construct the centroid for a right, obtuse, and acute triangle and write several sentences comparing
the location of the centroid.
b. Construct the orthocenter for a right, obtuse, and acute triangle and write several sentences
comparing the location of the orthocenter.
c. Construct the circumcenter for a right, obtuse, and acute triangle and write several sentences
comparing the location of the circumcenter.
d. Finally, construct the centroid, orthocenter, and circumcenter for an acute triangle. Then construct
the Euler Line. What does the Euler Line show is true for any triangle?
Grading Process
All drawings should be done neatly in pencil on graph paper. Pictures should fill the paper and be neatly
labeled. Research and comparisons should be double-spaced typed in 12 pt. font. This part should be about 1
to 2 pages in length. Any research should have proper citation as well. The project is an individual project.
All projects will be due on Friday, April 22nd . Don’t wait until the day before to do it!
The projects will be worth 100 points and will be graded based on the following rubric.
Research/Written Portion (40 points)





Typed, double-spaced (5 points) _________/5
1-2 pages in length (5 points) _________/5
Correct grammar/error-free (10 points) _________/10
Research cited correctly (5 points) _________/5
Relevant information (15 points) _________/15
o answers all questions in description
o addresses all important information about mathematician or mathematical idea
Construction (60 points)




Neat and appealing drawing (10 points) _________/10
Drawings done in pencil (5 points) _________/5
Drawings done on graph paper (5 points) _________/5
Accuracy of construction (20 points) _________/20
o objects are in correct locations
o objects are labeled correctly
o no unnecessary lines or points
 Completed all constructions (20 points) _________/20
Total: Grade_________ _________/100