[b][color=#000000]Theorems:[/color][/b][br]The center of a triangle's [b][color=#000000]9-Point Circle[/color][/b] is collinear with the triangle's[color=#38761d][b] centroid [/b][/color]and [b][color=#ff7700]orthocenter.[/color][/b][br](This segment is a subset [part] of the triangle's Euler Line: line that passes through the triangle's circumcenter, [color=#38761d][b]centroid[/b][/color], and [color=#ff7700][b]orthocenter.[/b][/color]) [br][br]Nonetheless, this applet also shows that the center of any triangle's 9-Point Circle lies [color=#ff00ff][b]1/4 of the way[/b][/color] along the segment that connects its [color=#38761d][b]centroid[/b][/color] to its [color=#ff7700][b]orthocenter[/b][/color]. In other words, the [b][color=#000000]ratio of[/color][/b] the distance from a triangle's [color=#38761d][b]centroid [/b][/color]to the center of its 9-Point Circle [b][color=#000000]to the[/color][/b] distance from a triangle's centroid to its orthocenter = [color=#ff00ff][b]1/4[/b][/color].