Concurrencies
The Circumcenter
Using the applet below, construct the perpendicular bisector of each segment of the triangle One such line (red), and how to construct it, are shown.
Use 2 circles with centers at 2 angles, and draw a segment from one intersection to the other.
If your diagram is getting too crowded with constructions, you may right-click on any construction and un-check the box "Show Object." This will hide the object from view, allowing you to only see what you need on the diagram.
Circumcenter
Question 1:
Using the Move tool (the mouse cursor icon), click on one of the triangle's angle points and move it around. Describe what happens to the intersection of your lines as you do so.
Question 2:
Construct a circle that passes through all three angle points of the triangle. What do you notice about the placement of the center of the circle?
The Centroid
Using the applet below, find the midpoint of each segment of the triangle and connect it to the opposite angle with another segment. One such segment (red), and how to construct it, are shown.
Use 2 circles with centers at 2 angles, and draw a segment from one intersection to the other. They will cross the side of the triangle at its midpoint.
Find the Centroid
Question 3:
Use the move tool again on one of the angle points on this diagram to move it around. Describe how the movement of this intersection is different from question 1.
The Incenter
In the triangle below, construct a circle inside the triangle such that the circle touches each segment of the triangle without crossing outside of it. Construct rays originating at each angle point that pass through the center of the circle.
Question 4:
Why do you think the intersection point is called the incenter?