**Assignment 4: **

**Centers of a Triangle**

**The purpose of this assignment is to
investigate the construction and properties of several different centers of a
triangle - the centroid, the orthocenter, and the circumcenter. Also, we
will investigate the relationship between the three centers.**

**Centroid**

__ Construction:__ The centroid is found by finding all
three midpoints of the segments that make up the triangle and connect each
midpoint to the corresponding vertex opposite each midpoint. After
completing all three lines from each midpoint to the opposite vertices, you can
see that all lines go through the same point in the center of the triangle.
This point is the intersection point of the lines. Once the intersection
point is constructed, it becomes the centroid of the triangle. Click
here to see the GSP construction of the centroid.

__ Properties:__ The centroid is one center of
the triangle, but it is not equal distance from each of the three vertices.
No matter which way you move or change the triangle that you have constructed
along with the centroid, the centroid is always in the center of the triangle.

**Circumcenter**

__ Construction:__ The circumcenter of a triangle
is found by finding the midpoints of the segments that comprise a triangle and
drawing the perpendicular bisectors of each of the three segments. After
completing tall the perpendicular bisectors of all three segments, you can see
that the perpendicular bisectors intersect each other at one point.
Further, this point is equal distance from all three vertices of the triangle.
By constructing a circle using the intersection point we have found as the
center and passing through one of the vertices of the original triangle (circle
by center + point), we now have a circle in which all the vertices of the
original triangle lie on the circle. Further, the center of the
constructed circle is the circumcenter. Click
here to see the GSP construction of the circumcenter.

__ Properties:__ When looking at the circumcenter
of an equilateral or acute triangle, the circumcenter is in the center of the
triangle (click here to
view the circumcenter of an acute and equilateral triangle). However, when
the triangle is obtuse, the circumcenter is outside of the triangle (click
here to view the circumcenter
of an obtuse triangle). When looking at a right triangle, the circumcenter
is on the midpoint of the hypotenuse of the triangle (click
here to view the circumcenter
of a right triangle). When moving a given triangle from an acute angle to
an obtuse angle, the circumcenter exits through the midpoint of the segment
opposite the obtuse angle.

**Orthocenter**

__ Construction:__ The orthocenter is found by
constructing a line perpendicular to a vertex and the segment of the triangle
that is opposite that vertex. This process is repeated for the other two
vertices and their corresponding opposite segments of the triangle. This will
construct three lines with a common intersection point. This intersection
point is the orthocenter. Click here to see
the GSP construction of the orthocenter.

__ Properties:__ When looking at a right
triangle, the orthocenter is at the vertex of the angle opposite the hypotenuse
(click here to see the
orthocenter of a right triangle). When looking at an equilateral triangle,
the orthocenter is in the center of the triangle (click here to see the orthocenter of
an equilateral triangle). When looking at a triangle with an acute angle,
the orthocenter is inside the triangle (click
here to see the orthocenter of an
acute triangle). When looking at a triangle with an obtuse angle, the
orthocenter exits the triangle through the vertex forming the obtuse angle and
then through the vertex point (click
here to see the orthocenter of
an obtuse angle).

**Observing All the Centers Together**

__ Construction:__ Click
here to see the GSP
construction of all the triangle centers on the same triangle.

__ Properties:__ All three of the centers are the
same when the triangle is equilateral (click
here to see the way all the
centers are exactly the same in an equilateral triangle). Also, the
centroid is always in the center of the orthocenter and the circumcenter no
matter how you change the triangle. When studying the distance between the
orthocenter and the circumcenter, the distance between the circumcenter and the
centroid is a ratio of 1/3 and the ratio of the distance between the orthocenter
and the centroid is 2/3.