Triangles: Points of Concurrency 2022
Circumcenter, Orthocenter, Centroid, Incenter, Perpendicular Bisectors, Altitudes, Medians, Angle Bisectors, and Euler's Line
Table of Contents
- Intro to Geogebra
- Introduction to GeoGebra
- Special Triangle Line Segments
- Triangle Altitudes 2022
- Perpendicular Bisectors 2022
- Angle Bisector 2022
- Triangle Medians 2022
- Triangle Centers Exploration
- Points of Concurrency 2022
- Euler's Line
- Exploring Points of Concurrency 2022
- Nine Point Circle
- The Nine Point Circle 2022
Introduction to GeoGebra
Play around with the toolbar to learn about the tools.
Triangle Altitudes 2022
What is a triangle altitude?
Interact with the applet for a few minutes.[br][br]Move the slider. [br] [br]The [color=#9900ff][b]purple segment[/b][/color] that will appear is said to be an [b][color=#9900ff]ALTITUDE OF A TRIANGLE.[/color][/b] [br]Be sure to move the [b][color=#1e84cc]blue vertex[/color][/b] and the white points of the triangle around a bit as well. [br][br]Then, answer the questions on your handout.
Points of Concurrency 2022
Centroid
1. Choose the "Polygon" tool and draw a large triangle.[br]2. Choose the "Point" tool and select Midpoint. Use this tool to locate and mark the midpoint of each side of the triangle.[br]3. Choose the "Line" tool and select Segment. Use this to draw a segment from each midpoint to the vertex across from it.[br]4. Use the "Point" tool to mark the intersection of all these segments.[br]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.][br]6. What do you notice about the lengths of these segments? [br]7. Does your observation hold true if you move the vertices of the triangle? Try several different triangles to find out![br]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.[br]2. Choose the "Construct" tool (fourth from the left) and select Angle Bisector.[br]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.][br]4. Use the "Point" tool to mark the intersection of your three lines.[br]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 [i]side[/i] of your triangle.[br]6. What do you notice about this circle? [br]7. What does that mean about the distance from each triangle side to the incenter? Use your measuring tool to confirm this.[br]8. On your paper, sketch a picture with above measurements included.
Incenter
Circumcenter
1. Draw a large triangle.[br]2. Choose the "Construct" tool and select Perpendicular Bisector. [br]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.][br]4. Use the "Point" tool to mark the intersection of your three lines.[br]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 [i]vertex[/i] of your triangle. [If your triangle is too big, you can zoom out by choosing the last tool on the row.][br]6. What do you notice about this circle? [br]7. What does that mean about the distance from each vertex to the circumcenter? Use your measuring tool to confirm this.[br]8. On your paper, sketch a picture with your measurements.
Circumcenter
Orthocenter
1. Draw a large triangle.[br]2. Choose the "Construct" tool and select Perpendicular Line. [br]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.][br]4. Use the "Point" tool to mark the intersection of your three lines.[br]5. Is it an equal distance from this point to each side? To each vertex? Measure each segment. Is there a common theme?[br]6. On your paper, sketch a picture and fill in the blanks.
Orthocenter
Exploring Points of Concurrency 2022
What is Euler's line?
Manipulate the vertices of the triangle to investigate the location of the centers of a triangle.
The Nine Point Circle 2022
Construct the Nine Point Circle, as shown and described below; must pass the "drag test."[br][br]1. In triangle ABC construct the midpoints of the three sides. Right click on the points and "show label". Midpoints should be labeled D, E, and F.[br][br]2. In triangle ABC construct the altitudes of the three sides. Mark and label (right click) the orthocenter "O". [br][br]3. Mark the base of the altitudes where each line meets the sides of the triangle. Label these points G, H, J. You will need to rename point I to J. [br][br]3. In triangle ABC construct the midpoints of the distance between the orthocenter and each of the vertices. Label these points K, L, and M[br][br]4. In triangle ABC construct the circumcenter. Label this point Ci.[br][br]5. In triangle ABC construct the midpoint between the circumcenter and the orthocenter. Label this point Mid.[br][br]6. Construct a circle with its center at Mid and a point on the circumference at the midpoint of one of the sides of triangle ABC; this is the Nine Point Circle