Points of Concurrency 2022
Centroid
1. Choose the "Polygon" tool and draw a large triangle.
2. Choose the "Point" tool and select Midpoint. Use this tool to locate and mark the midpoint of each side of the triangle.
3. Choose the "Line" tool and select Segment. Use this to draw a segment from each midpoint to the vertex across from it.
4. Use the "Point" tool to mark the intersection of all these segments.
5. Choose the "Measure" tool (fourth from the right) and select Distance or Length. Measure the length of all six segments that are inside your triangle. [Each segment you drew earlier is now broken into two segments.]
6. What do you notice about the lengths of these segments?
7. Does your observation hold true if you move the vertices of the triangle? Try several different triangles to find out!
8. On your paper, sketch a picture of your triangle with the inside segments drawn. Include your measurements on your sketch.
Centroid
Incenter
1. Draw a large triangle.
2. Choose the "Construct" tool (fourth from the left) and select Angle Bisector.
3. Use the tool to create an angle bisector at each vertex of the triangle. [To do this, select three points in a row. The middle selection is where your line will be drawn.]
4. Use the "Point" tool to mark the intersection of your three lines.
5. Choose the "Circle" tool and select Circle with Center through Point. Use it to draw a circle whose center is your Incenter and which touches at least one side of your triangle.
6. What do you notice about this circle?
7. What does that mean about the distance from each triangle side to the incenter? Use your measuring tool to confirm this.
8. On your paper, sketch a picture with above measurements included.
Incenter
Circumcenter
1. Draw a large triangle.
2. Choose the "Construct" tool and select Perpendicular Bisector.
3. Use the tool to create a perpendicular bisector on each side of the triangle. [To do this, select the two points at the end of a side. Repeat for each side.]
4. Use the "Point" tool to mark the intersection of your three lines.
5. Choose the "Circle" tool and select Circle with Center through Point. Use it to draw a circle whose center is your Circumcenter and which touches at least one vertex of your triangle. [If your triangle is too big, you can zoom out by choosing the last tool on the row.]
6. What do you notice about this circle?
7. What does that mean about the distance from each vertex to the circumcenter? Use your measuring tool to confirm this.
8. On your paper, sketch a picture with your measurements.
Circumcenter
Orthocenter
1. Draw a large triangle.
2. Choose the "Construct" tool and select Perpendicular Line.
3. Use the tool to create an altitude through each vertex that is perpendicular to the opposite side. [To do this, first select the vertex then select the side opposite from that vertex. Repeat for each vertex.]
4. Use the "Point" tool to mark the intersection of your three lines.
5. Is it an equal distance from this point to each side? To each vertex? Measure each segment. Is there a common theme?
6. On your paper, sketch a picture and fill in the blanks.