 EMAT 6690

~Circumcenter~Orthocenter~

~Euler Line ~

A perpendicular bisector is a line that bisects and is perpendicular to the side of a triangle.  The point of concurrency of the perpendicular bisectors is called the circumcenter of a triangle.

An investigation than can be used as a teaching tool for the circumcenter.

Investigation

• Step 1    On a sheet of paper or wax paper, draw a large scalene acute triangle.  Make the triangle as large as you can.
• Step 2     Take the acute triangle that you constructed and label it ABD, as shown below. • Step 3    Construct the three perpendicular bisectors.  Find the midpoints of the three sides of the triangle, and label midpoints E, F, and J.  Label the circumcenter C. • Step 4    Use your compass to answer the following questions:  Is the circumcenter the same distance from the vertices?  From the three sides?  Take a look at how the circumcenter divides each perpendicular bisectors.  Is the circumcenter the midpoint of each perpendicular bisector?
• Step 5    The circumcenter is the same distance from the vertices, but it is not the same distance from the three sides.  The circumcenter is not the midpoint of each perpendicular bisector.

An altitude is a perpendicular line drawn from a vertex of a triangle to the opposite side.  The point of concurrency of the altitudes is called the orthocenter of a triangle.

An investigation than can be used as a teaching tool for the circumcenter.

Investigation

• Step 1    On a sheet of paper or wax paper, draw a large scalene acute triangle.  Make the triangle as large as you can.
• Step 2     Take the acute triangle that you constructed and label it ABD, as shown below. • Step 3    Construct the three altitudes, and label the altitudes AE, BF, and DJ.  Label the orthocenter H. • Step 4    Use your compass to answer the following questions:  Is the orthocenter the same distance from the vertices?  From the three sides?  Take a look at how the orthocenter divides each perpendicular bisectors.  Is the orthocenter the midpoint of each perpendicular bisector?
• Step 5    The orthocenter is not the same distance from the vertices and the three sides.  The orthocenter is not the midpoint of each perpendicular bisector.

The Euler Line is always contains three of the four points of concurrency.  THe Euler Line is named after Swiss mathematician Leonhard Euler, who proved that the three points of concurrency were collinear.

An investigation that can be used as a teaching tool for the Euler Line.

Investigation

• Step 1    On a sheet of paper or wax paper, draw a large scalene acute triangle.  Make the triangle as large as you can.
• Step 2    Take the acute triangle and label it ABD, as shown below. • Step 3     Now construct the four points of concurrency on your triangle.  Label the centroid G, the incenter I, the circumcenter C, and the orthocenter H. Students should be able to complete the following conjecture at this point.

The -(centroid)-,-(circumcenter)-, and the -(orthocenter)- are the three points of concurrency that always lie on the Euler line.

• Step 4    With a compass, compare lengths of the two parts on the Euler segment.

Students should be able to complete the following conjecture at this point.

The -(centroid)- divides the Euler segment into two parts so that the smaller part is -(half as long as)- the larger part.

Student Activity

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