"New" Ways of Working with Triangles 

Hans-Georg Weigand and James P. Dildine

The following links allow an exploration into new ways of working with triangles that can be afforded to us through technology.

Working with flexible triangles:

We want you to explore the locus of special points of a triangle.

We solve problems in five steps

Circumcenter

 

Center of gravity

 

Orthocenter

 

Go ahead with the following problems. Try to solve them first by your own. Afterward you will see the solution. 

First: Given is the triangle ABC shown below (The point C should be on the parallel!). Point C moves along a line that is parallel to the line segment AB and the line the segment is contained in.

Problem 1: First step: You see the CIRCUMCENTER of a triangle ABC. If you move point C along the line parallel to base AC... What locus of points is created if C moves along the line parallel to the base. Do a hand draft of this locus.

 

Second step: PICTURE were you can do a hand draft.

 

Third step: Next PICTURE (without perpendicular bisectors): Now you can move the point C. Also animate .....

 

Forth step: Why do you think that the formation created is as such? Write it down: TEXTBOX.

 

Fifth step: If you don't know it, you may also look at the next PICTURE (with perpendicular bisectors): Do you have an idea now, why. ...

 

????There is a possibility to go on e. g.

Problem 2: Now point C moves along a circle (or on another curve). ......

.........

Problem 3: Now we consider a quadrilateral (4-sided polygon) ABCD. Construct the perpendicular bisectors of the quadrilateral ABCD

This is a completely different problem....But this can lead to the classification of quadrilaterals.

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The CIRCUMCENTER of a triangle results at the intersection of the perpendicualr bisectors of he sides of triangle ABC. You can construct the cirumcenter .....Back

 

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Problem II: You see CENTER OF GRAVITY of a triangle ABC. If you move point C along the line parallel to base AC... What locus of points is created if C moves along the line parallel to the base. Do a hand draft of this locus.

 

PICTURE were you can do a hand draft.

 Next PICTURE (without perpendicular bisectors): Now you can move the point C. Also animate .....

 Why do you think that the formation created is as such? Write it down: TEXTBOX.

If you don't know it: Do you know any property concerning the bisectors and the center of gravity.

Do you know what a dialation is? Now we do a dialation......

 

Problem 2: Now we consider a quadrilateral (4-sided polygon) ABCD. Construct the perpendicular bisectors of the quadrilateral ABCD

This is a completely different problem....But this can lead to the classification of quadrilaterals.

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The five steps:

First step: The problem

Second step: Enactive level: Do it by hand

Third step: Do a computer animation

Forth step: Explain the computer picture

Fifth step: The computer as a help for explaining the problem.

Back

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The CIRCUMCENTER of a triangle results at the intersection of the perpendicualr bisectors of the sides of triangle ABC.

You can construct the cirumcenter .....

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The CENTER OF GRAVITY OF A TRIANGLE is the intersection of the lines that make up the bisectors of sides triangle ABC.

If you move point B along the line parallel to the base AC then...What locus of points does this intersection point create? Draw a

picture on your own first and then use the interactive simulator here.

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Circumcenter Exploration

Center of Gravity (Median Center)

Orthocenter Exploration

 


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