Mathematics Education
This Assignment has a set of activities to help become familiar with GSP and to review some basic triangle geometry. After examining the activitites in the assignment, pick some topic for a brief write-up. The write-up could be one of the proofs, but does not have to be. It could be some exploration you would try with students. Or you might take one of the topics (e.g. medians) and explore some of the standard geometry for the topic.
1. The CENTROID (G) of a triangle is the common intersection of the three medians. A median of a triangle is the segment from a vertex to the midpoint of the opposite side.
Use Geometer's Sketchpad (GSP) to Construct the centroid and explore its location for various shapes of triangles.
2. The ORTHOCENTER (H) of a
triangle is the common intersection of the three lines containing
the altitudes. An altitude is a perpendicular segment from a vertex
to the line of the opposite side. (Note: the foot of the perpendicular
may be on the extension of the side of the triangle.) It should
be clear that H does not have to be on the segments that are the
altitudes. Rather, H lies on the lines extended along the altitudes.
Use GSP to construct an orthocenter H and explore its location for various shapes of triangles. (Make sure your construction holds for obtuse triangles.)
3. The CIRCUMCENTER (C) of a
triangle is the point in the plane equidistant from the three
vertices of the triangle. Since a point equidistant from two points
lies on the perpendicular bisector of the segment determined by
the two points, C is on the perpendicular bisector of each side
of the triangle. Note: C may be outside of the triangle.
Construct the circumcenter C and explore its location for various shapes of triangles. It is the center of the CIRCUMCIRCLE (the circumscribed circle) of the triangle.
4. The INCENTER (I) of a triangle
is the point on the interior of the triangle that is equidistant
from the three sides. Since a point interior to an angle that
is equidistant from the two sides of the angle lies on the angle
bisector, then I must be on the angle bisector of each angle of
the triangle.
Use GSP to find a construction of the incenter I and explore its locationfor various shapes of triangles. The incenter is the center of the INCIRCLE (the inscribed circle) of the triangle.
5. Use GSP to construct G, H,
C, and I for the same triangle. What relationships can you find
among G, H, C, and I or subsets of them? Explore for many shapes
of triangles.
6. Take any triangle. Construct
a triangle connecting the three midpoints of the sides. This is
called the MEDIAL triangle. It is similar to the original triangle
and one-fourth of its area. Construct G, H, C, and I for this
new triangle. Compare to G, H, C, and I in the original triangle.
7. Take any acute triangle.
Construct a triangle connecting the feet of the altitudes. This
is called the ORTHIC triangle. Construct G, H, C, and I for the
orthic triangle. Compare to G, H, C, and I in the original triangle.
Can you extend this to right triangles or obtuse triangles?
8. Take an acute triangle ABC.
Construct H and the segments HA, HB, and HC. Construct the midpoints
of HA, HB, and HC. Connect the midpoints to form a triangle. Prove
that this triangle is similar to triangle ABC and congruent to
the medial triangle. Construct G, H, C, and I for this triangle.
Compare.
9. In the same original triangle,
construct the three secondary triangles of Exercises 6, 7, and
8. Construct the circumcircle for each of the secondary triangles.
What do you observe? Can you prove your conjecture?
10. The Nine-Point circle for
any triangle passes through the three mid-points of the sides,
the three feet of the altitudes, and the three mid-points of the
segments from the respective vertices to orthocenter. Construct
the nine points, locate the center (N) and construct the nine
point circle.
11. How is N related to G, H,
C, or I for different shaped triangles?
12. Prove that the three perpendicular
bisectors of the sides of a triangle are concurrent.
13. Prove that the lines of
the three altitudes of a triangle are concurrent.
14. Prove that the three medians
of a triangle are concurrent and that the point of concurrence,
the centroid, is two-thirds the distance from each vertex to the
opposite side.
How would you use GSP to help students understand this relationship of the triangle and its medians? How would you develop a sense of proof of the relationship with students?
15. Prove that the three angle
bisectors of the internal angles of a triangle are concurrent.
16. Prove that any angle bisector
of a triangle is concurrent with the two angle bisectors of the
opposite exterior angles of a triangle.
17. Take a point of concurrence
as determined in Problem 16 and construct a circle tangent to
the lines of the three sides (of the triangle)
18. Prove that for any triangle,
H, G, and C are collinear, and prove that HG = 2GC.
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