Ashley Estes
EdSec 426
Jennifer Cribbs
History and Technology Component
What is triangle geometry? According to Encyklopadie authors, it boils down to an
arbitrary triangle, with certain points (and lines and curves), which are then determined to have
remarkable properties with respect to a triangle (Davis). The triangle seemed to have been first
discovered independently by both the Persians and the Chinese way back in the eleventh century.
It was used to “extract square and cube roots of numbers” by a Chinese mathematician named
Chia Hsien (Clawson). Also having a way of extracting roots of numbers, a Persian
mathematician by the name of Omar Khayyam, had knowledge of what was called “Pascal’s
Triangle.” However, Yang Hui, another Chinese mathematician, seemed to have the earliest
display of Pascal’s Triangle in 1301. Pascal’s Triangle was named after Blaise Pascal; he
developed many of the triangle’s properties and applications (Eves). Triangle Geometry seemed
to have emerged in the 1870’s in the writings of E. Lemoine. From that point on, many
mathematicians have contributed to the development of the triangle since then. The nine point
circle theorem is one of the most known results that came from the triangle. It says that the
following nine points are concyclic: the three side bisectors, the three altitude feet, and the three
midpoints along the altitudes from the vertices to the orthocenter. It is not one hundred percent
clear, of how triangle geometry discovered, except that there was a lot “playing around” before
anything came about. A triangle can be drawn using a ruler and compass, but now we have
technology available for us, like computers, that are able to do just that. However, triangle
geometry has died off. This is due to the fact that no new ideas seemed to emerge from triangle
geometry. However, one major idea that did emerge from triangle geometry is geometrography,
which is as follows: beginning with a simple figure (often a triangle), construct a distinguished
point or figure, often with a ruler and compass, but also with other means, and then count how
many elementary operations required to do so (Davis).
With that being said, since the origination of the triangle and triangle geometry, it has
been applied to theorem discovery, mathematical experimentation, and the use of computers. The
use of computers is huge because computers are part of everyday life for most of us, especially
when it comes to students’ lives. Students are surrounded by technology, which is why I feel as
though it is important to incorporate technology into our teaching, such as the use of computers.
One great way to incorporate the use of a computer in teaching students about the properties of
triangles is by using Geometer’s sketchpad. Geometer’s sketchpad is not only easy to use, but
has a variety of different tools that can be used to help students better understand properties of
triangles, perimeter, area, etc. In order to facilitate learning through Geometer’s sketchpad, I
think it is worthwhile to take the time to show students how to use the program and allow them
to get familiar with it in order to work up to answering problems using the program. In order to
start, the students can learn how to draw a right triangle, an equilateral triangle, etc. This way
they can get use to using the different tools and where each tool is in the program. Once the
students are familiar with the technology, problems could be distributed in which there are
simple, straight forward problems as well as word problems. The word problems can be very
worthwhile for the students because the question can be asking more than one thing and could be
requiring the students to draw more than one shape in order to answer the question.
I think Geometer’s Sketchpad is a great way for students to learn more about triangle
properties because the program offers so much and it is a great way to engage the students.
Through this program, students are allowed to work independently and must know certain
properties of triangles in order to make the triangle itself. For example: a student must know that
a right triangle is 90 degrees because they must form a 90 degree angle in order to make the
triangle using Geometer’s Sketchpad. Although Geometer’s Sketchpad may draw the lines for
the students and connect the dots, the students do have to have previous knowledge about the
subject going into use the program, which is why I like it because it requires them to still think
about what they are doing rather than the program doing all the work for themselves. This way,
Geometer’s Sketchpad can facilitate student thinking, while engaging them at the same time.
Another type of technology that can be used that is quite similar to Geometer’s Sketchpad is a
program by the name of Geogebra. While the school may provide Geometer’s Sketchpad on
computers, it does cost money if the student wanted to download the program on a computer at
home. Geogebra is free and can be accessed at home if a student has a computer with internet.
Geogebra can do much of what Geometer’s Sketchpad can, which means that students can use
this program outside of the classroom to further their learning in triangle geometry. Visualizing
is such a huge part of geometry, which is why these programs are so beneficial for students.
These programs allow students to put what they are learning into pictures; pictures which are
worth a million words.
Works Cited
Davis, Philip. The Rise, Fall, and Possible Transfiguration of Triangle Geometry: A Mini
History. 3. 102. Providence, RI: Division of Applied Mathematics, 1995. 204-214. eBook.