[color=#000000]For every triangle that exists, there is a[/color] [color=#1e84cc][b]very special circle[/b][/color] that passes through [color=#ff00ff][b]9 points[/b][/color]. [color=#000000]Some of these [/color][color=#ff00ff][b]points[/b][/color][color=#000000] lie on the triangle itself, and some do not. [/color][br][br][color=#000000]The applet below will informally illustrate the construction of a triangle's[/color] [color=#1e84cc][b]9-point circle[/b][/color][color=#000000]. [br]Be sure to change the locations of the triangle's[/color][b] BIG WHITE VERTICES[/b][color=#000000] each time before re-sliding the slider. It would also be wise to alter the locations of these [/color][b]vertices[/b][color=#000000] [b][i]after[/i][/b] you've constructed this [/color][color=#ff00ff][b]9-point circle[/b][/color][color=#000000]. [br][br][/color][i][color=#ff0000]Take your time with this applet![/color][/i][color=#000000] Study its dynamics very carefully. Answer the questions that follow. [br][br][/color]

[color=#000000][b]Questions: [/b][/color] [br][br][color=#000000]1) Where exactly is the [/color][color=#1e84cc][b]center[/b][/color][color=#000000] of a triangle's [/color][color=#1e84cc][b]9-point circle[/b][/color][color=#000000] located?[br] That is, how would you describe to somebody how to locate it? [br] Be sure to use appropriate vocabulary terms and be sure to be specific in your response! [br][br]2) Describe the [/color][color=#ff00ff][b]points[/b][/color][color=#000000] that are located on a triangle's [/color][color=#1e84cc][b]9-point circle[/b][/color][color=#000000]. What exactly are these [/color][color=#ff00ff][b]points[/b][/color][color=#000000]? [br] That is, how do these [/color][color=#ff00ff][b]points[/b][/color][color=#000000] relate to features of the triangle itself? [br][br]3) Does the [/color][color=#1e84cc][b]center[/b][/color][color=#000000] of a triangle's [/color][color=#1e84cc][b]9-point circle [/b][/color][color=#000000]always lie inside the triangle? [br] [br]4) Is it ever possible for any [/color][color=#ff00ff][b]2 or more[/b][/color][color=#000000] of these [/color][color=#ff00ff][b]9 points[/b][/color][color=#000000] to overlap? That is, did you observe any cases [br] where [/color][color=#ff00ff][b]2 (or more) pink points coincide [/b](lie on top of each other)[/color][color=#000000]? If so, describe any possible [br] conditions/features of the triangle for which this behavior occurred.[br][/color][i][color=#1e84cc] Be sure to use the tools of GeoGebra to help you provide answer(s) to this question! [/color][/i]