[color=#000000][b]9-Point Circle Theorem:[/b][/color][br]For any triangle, there are [color=#ff00ff][b]9 special points[/b][/color] that [color=#ff00ff][b]all lie[/b][/color] on the [color=#0000ff][b]one circle[/b][/color]. These points are:[br]-The [color=#ff00ff][b]midpoints of the triangle's 3 sides[/b][/color][br]-The [color=#ff00ff][b]points[/b][/color] at which the lines containing the triangle's 3 altitudes intersect the lines that contain the triangle's 3 sides[br]-The [color=#ff00ff][b]midpoint [/b][/color]of each segment connecting the triangle's orthocenter to one of its vertices. (There are [color=#ff00ff][b]3 such points[/b][/color].) [br][br]The [color=#0000ff][b]center[/b][/color] of this [color=#0000ff][b]9-point circle[/b][/color] is the [color=#0000ff][b]midpoint[/b][/color] of the [color=#9900ff][b]segment[/b][/color] whose endpoints are the triangle's [color=#980000][b]circumcenter[/b][/color] and [color=#ff7700][b]orthocenter.[/b][/color]