Circumcenter
A [color=#ff0000][b]circumcenter[/b][/color] is the point of concurrency of the three perpendicular bisectors. [br]-Construct triangle XYZ and label the vertices with text tool[icon]/images/ggb/toolbar/mode_text.png[/icon].[br]-Construct the perpendicular bisector [icon]/images/ggb/toolbar/mode_linebisector.png[/icon]of each side.[br]-Use the intersect [icon]/images/ggb/toolbar/mode_intersect.png[/icon]in the point menu to mark the circumcenter and name it A with text tool [icon]/images/ggb/toolbar/mode_text.png[/icon].
Construct the circumcenter below.
Drag the vertices of XYZ around. What kind of triangle is XYZ if the circumcenter A falls on the exterior of the triangle?
What kind of triangle is XYZ if the circumcenter A falls on the triangle?
What kind of triangle is XYZ if the circumcenter A is in the interior of the triangle?
A [b][color=#9900ff]circumscribed circle[/color][/b] (circle that goes through each vertex) can be added in this construction above. Draw a segment from the [b][color=#ff0000]circumcenter[/color][/b] A to vertex X. This creates the radius of the circle (segment AX). Construct a circle on your construction above using your compass tool[icon]/images/ggb/toolbar/mode_compasses.png[/icon]. Move the triangle around and verify the circle always goes through all vertices X, Y and Z.
Drag around the vertices of XYZ. Does the circle always go through the three vertices and remain outside the triangle?
Since a circumscribed circle goes through each vertex, the circumcenter is equidistant from each:[br][br]