The Geometer's Sketchpad gives the user the facility to analyze many
different aspects of triangles that become obvious with this tool. Doing
the same constructions with compass and straightedge would be time consuming
and without the accuracy would not demonstrate the concepts as clearly.
In my class with 7th graders, we have constructed the incenter with straightedge
and compass with some degree of success; however the advantages of the GSP
construction are obvious.
To begin the study of the centers of a triangle, we start with the centroid
by constructing the medians, with connecting the vertices to the opposite
midpoints.
On GSP we can move any vertex to see the movement of the centroid with
the changes in the triangle.
The centroid does move but not dramatically as the triangle is changed
from acute to right and to obtuse.
The second triangle construction is the orthocenter. The orthocenter is
the point of intersection of the altitudes of the triangle.
When the triangle is changed to right and obtuse the orthocenter moves
radically.
When the triangle is right, H is on the vertex of the right angle, and
with an obtuse triangle, the orthocenter is outside the triangle.
The circumcenter is the third construction. The circumcenter is equidistant
from the three vertices and therefore is the center of a circle that circumscribes
the triangle. To find that point, construct the perpendicular bisector of
each side. The circumcenter is the center of the circmcircle as shown ineach
of the following.
Once again, we change the triangle to see the effect on the circumcenter.
On a right triangle the circumcenter is on the hypotenuse, and on an
obtuse triangle it is outside the triangle.
The fourth center is the incenter, constructed by bisecting the angles
to find points equidistant from the sides. The incenter is the center of
the inscribed circle of the triangle.
Changing the shape of the triangle produces the following changes in
the incenter.
When G,H and C are all constructed in one triangle, they are collinear.
This line is Euler's line, shown here in blue. Changing the triangles demonstrates
some of the qualities of Euler's line; for example, in the right triangle
it is the median.
The following show the consistency of Euler's line.
When the Medial triangle is constructed by connecting the midpoints of the
triangle and the G, H and C are constructed for the new triangle the line
they coincide with the ones for the original.
The following triangle has the Medial triangle in green, formed by connecting
the midpoints of the original triangle. The incircle in red, and Euler's
line in blue.
The Nine-point circle can be constructed by finding midpoints of the sides,
points at the feet of the altitudes and midpoints of the segments from orthocenter
to the respective vertices. In order to find the center of the nine-point
circle, use the segments for diameters, the intersection gives the center
and a radius for the circle.
There are many more investigations of triangles in Assignment 5, where scripts
are written to repeat these constructions automatically.
Proof:
Given: any triangle, with 3 perpendicular bisectors.
To Prove: The 3 perpendicular bisectors are concurrent.
