In your own words, what is the center of a shape? How do you find the center of a circle or square? Think about how you would you go about finding the center of a triangle.

The center of a triangle can be defined in 4 main ways. These are[br][br][list][*]Circumcenter[/*][*]Orthocenter[/*][*]Centroid[/*][*]Incenter[/*][/list][br]By interacting with the diagram above, explore each one of these centers. By clicking "construction" you can see how these centers can be defined and obtained. Write a description in your own words of how each one of these centers is defined.[br][br]HINT: It is best to only show one center and the construction lines at a time[br][br]HINT: Pay close attention to the markings that appear on the construction lines/angles. What do they mean?

Using only a ruler and a straight-edge, describe how you would go about constructing each of the above triangle centers.[br][br]In addition to the drawing of lines, segments and circles, you will need to refer to some of the fundamental construction techniques. [br][list][*][b]Construction 1:[/b] Construct the perpendicular bisector of a line segment.[/*][*][b]Construction 2:[/b] Construct the perpendicular line from a given point to a line not containing that point.[/*][*][b]Construction 3:[/b] Construct a line perpendicular to a given line at a given point on that line.[/*][*][b]Construction 4:[/b] Construct a line that bisects an arbitrary angle.[/*][/list]What does each one require and what does it produce?

Turn one or more of the triangle centers on (and the corresponding construction lines if you wish). Use the mouse to drag one of the corners of the triangle around.[br][br]As you do this, what do you notice about the triangle centers?[br][br]In particular you may want to pay attention to:[list][*]Do they ever overlap? If so, when?[/*][*]Are all of them always "inside" the triangle?[/*][*]Where are the centers located for different types of triangle?[/*][*]Do they all move in the same way? Is there any pattern that you notice about the way they move?[/*][/list]