3 Special Points!
Recall the following:
1) The lines that contain a triangle's 3 altitudes are concurrent (intersect at exactly one point.)
This point of concurrency is called the orthocenter of the triangle.
2) A triangle's 3 perpendicular bisectors are concurrent at a point called the circumcenter of the triangle.
3) A triangle's 3 medians are concurrent at a point called the centroid of the triangle.
Interact with the applet below for a few minutes. Then answer the discussion questions that follow.
Questions:
1) What conclusion can you make about the positioning of a triangle's orthocenter, circumcenter, and
centroid? Explain how you can use the toolbar to illustrate this.
2) How does the sliding the slider also informally show that your response to (1) is true?
3) Let's denote the orthocenter as O, the circumcenter as C, and the centroid as G.
What is the exact value of the ratio CG/CO? What is the exact value of the ratio CG/GO?
4) Prove your assertion for (1) true using a coordinate geometry format.
For simplicity's sake, position the triangle so its vertices have coordinates (0,0), (6a, 0), and (6b, 6c).
5) Prove your responses to (3) are true using the same coordinate geometry setup you used in (4) above.
6) Research information about the Euler Line of a triangle.
How does the Euler Line relate to the context of the above applet?