The point where a triangle's three [b][color=#ff0000]medians [/color][/b]intersect is called the [b][color=#0000ff]centroid[/color][/b]. A [b][color=#ff0000]median [/color][/b]of a triangle is a segment connecting a vertex to the midpoint of its opposite side. [br][br][i]To construct the centroid, it is sufficient to find the intersection of two medians, since the third median will also pass through this point.[/i][br][br][b][size=150]Construct a Centroid using MIDPOINT [icon]/images/ggb/toolbar/mode_mirroratpoint.png[/icon] tool[/size][/b][br][br]Follow the steps below to construct the centroid.[br][br][list=1][*]Construct the midpoint of side AB. Click on [icon]/images/ggb/toolbar/mode_mirroratpoint.png[/icon] and then on A and B. [b]*Repeat this step with side AC.[/b][br][/*][*]Construct a [b][color=#ff0000]median[/color][/b] on segment AB: Click on [icon]/images/ggb/toolbar/mode_segment.png[/icon] to connect the midpoint of AB with point C.[/*][*]Construct a [b][color=#ff0000]median[/color][/b] on segment AC. Click on [icon]/images/ggb/toolbar/mode_segment.png[/icon] to connect the midpoint of AC with point B.[/*][*]Mark the [b][color=#0000ff]centroid [/color][/b](the point of intersection of the medians): Click on [icon]/images/ggb/toolbar/mode_intersect.png[/icon] and then click on each median. [br][/*][/list]

[b][size=150]Construct a Centroid using COMPASS [icon]/images/ggb/toolbar/mode_compasses.png[/icon] tool[/size][/b][br][br]1) Construct the midpoint of side AB.[br][br][list][*]Construct a circle with radius AB centered on A. Click on [icon]/images/ggb/toolbar/mode_compasses.png[/icon] and then on point A and point B. Then, click on point A again to center on point A.[/*][/list][list][*]Construct a circle with radius AB centered on B. Click on [icon]/images/ggb/toolbar/mode_compasses.png[/icon] and then on point A and point B. Then, click on point B again to center on point B.[/*][/list][list][*]Mark the points of intersection of the two circles. Click on [icon]/images/ggb/toolbar/mode_intersect.png[/icon] and then click on each circle.[/*][/list][list][*]Construct a segment (perpendicular bisector) between the two points of intersection of the circles. Click on [icon]/images/ggb/toolbar/mode_segment.png[/icon] and then click on the two points of intersection.[/*][*]Construct the midpoint of side AB. Click on [icon]/images/ggb/toolbar/mode_intersect.png[/icon] and then click on the perpendicular bisector segment and then click on side AB of the triangle.[/*][*][i]Hide the circles, segment, and points of intersection of the circles. Click on [icon]/images/ggb/toolbar/mode_showhideobject.png[/icon] and then on any part that you want to hide. *Be sure not to hide the original triangle or the midpoints of the sides of the triangles.[/i][/*][*][b]Repeat these steps with side AC.[/b][br][/*][/list]2) Construct a [b][color=#ff0000]median[/color][/b] on segment AB: Click on [icon]https://www.geogebra.org/images/ggb/toolbar/mode_segment.png[/icon] to connect the midpoint of AB with point C.[br]3) Construct a [b][color=#ff0000]median[/color][/b] on segment AC. Click on [icon]https://www.geogebra.org/images/ggb/toolbar/mode_segment.png[/icon] to connect the midpoint of AC with point B.[br]4) Mark the [b][color=#0000ff]centroid [/color][/b](the point of intersection of the medians): Click on [icon]https://www.geogebra.org/images/ggb/toolbar/mode_intersect.png[/icon] and then click on each median.