[color=#000000]Recall the following: [/color][br][br][color=#ff7700]1) The lines that contain a triangle's 3 altitudes are concurrent (intersect at exactly one point.) [br] This point of concurrency is called the orthocenter of the triangle. [/color][br][br][color=#cc0000]2) A triangle's 3 perpendicular bisectors are concurrent at a point called the circumcenter of the triangle.[/color][br][br][color=#38761d]3) A triangle's 3 medians are concurrent at a point called the centroid of the triangle. [br][br][/color][color=#000000]Interact with the applet below for a few minutes. Then answer the discussion questions that follow. [/color]

[color=#000000][b]Questions: [/b][br][br]1) What conclusion can you make about the positioning of a triangle's[/color] [color=#ff7700]orthocenter,[/color] [color=#cc0000]circumcenter,[/color] [color=#000000]and[/color] [br] [color=#38761d]centroid[/color][color=#000000]? Explain how you can use the toolbar to illustrate this. [/color][br][br][color=#000000]2) How does the sliding the slider also informally show that your response to (1) is true? [/color][br][br][color=#000000]3) Let's denote the [/color][color=#ff7700]orthocenter as [/color][i][color=#ff7700]O[/color], [/i][color=#000000]the [/color][color=#cc0000]circumcenter as [i]C[/i][/color], [color=#000000]and the[/color] [color=#38761d]centroid as [i]G[/i][/color][color=#000000].[/color] [br] [color=#000000]What is the exact value of the ratio [i]CG/CO[/i]? What is the exact value of the ratio [i]CG[/i]/[i]GO[/i]?[/color] [br][br][color=#000000]4) Prove your assertion for (1) true using a coordinate geometry format. [br] For simplicity's sake, position the triangle so its vertices have coordinates (0,0), (6a, 0), and (6b, 6c). [br][br]5) Prove your responses to (3) are true using the same coordinate geometry setup you used in (4) above. [br][br]6) Research information about the Euler Line of a triangle. [br] How does the Euler Line relate to the context of the above applet? [/color]