Geometry 2nd Semester 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 part of 5% 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. Carefully follow the steps below to construct the Nine-Point Circle on graph
paper. Hint: Erase unnecessary lines as you go to prevent confusion
.
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.
Modified from: http://home.cvc.org/math/dgeom/Projects/Project.pdf
Option 2: Sierpinski Triangles
The Sierpinski Triangle is a fractal, a figure created by repeating the same process over
and over again. Do some 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 page
381 #9 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 page 278 #33-36 of
your textbook for more information)
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?
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 page in length. Any research should have
proper citation as well. The project is an individual project.
All projects will be due on Friday, January 11, 2013. Don’t wait until January 10 to do it!
The projects will be worth 100 points and will be graded based on the following rubric.
Modified from: http://home.cvc.org/math/dgeom/Projects/Project.pdf
Research/Written Portion (40 points)
-Typed, double-spaced (5 points)
_________/5
-1 page in length (5 points)
_________/5
-Correct grammar/error-free (10 points)
_________/10
-Research cited correctly (5 points)
_________/5
-Relevant information (15 points)
_________/15
-answers all questions in description
-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
-objects are in correct locations
-objects are labeled correctly
-no unnecessary lines or points
-Completed all constructions (20 points)
_________/20
Total: Grade
_________/100
Modified from: http://home.cvc.org/math/dgeom/Projects/Project.pdf