[b]Step 1: [/b] Use [icon]/images/ggb/toolbar/mode_angularbisector.png[/icon] to construct the angle bisectors of angles A, B and C.[br][b]Step 2:[/b] Use [icon]/images/ggb/toolbar/mode_intersect.png[/icon] to add a point where the three angle bisectors intersect. [br][b]Step 3:[/b] Use [icon]/images/ggb/toolbar/mode_showhidelabel.png[/icon] to label the point where the angle bisectors intersect.[br]
[b][color=#cc0000]The point where all three angle bisectors intersect is called the incenter. [br][/color][/b][br]1. No matter how you move the triangle, the [b][color=#900000]incenter[/color][/b] is always inside the triangle. Use [icon]https://www.geogebra.org/images/ggb/toolbar/mode_move.png[/icon] to adjust the triangle. What do you notice about vertex A if it is very close to the [b][color=#900000]incenter[/color][/b]?
2. Use [icon]/images/ggb/toolbar/mode_move.png[/icon] to adjust the triangle so that one vertex is definitely farther from the [b][color=#900000]incenter[/color][/b] than the other vertices. What do you notice about the angle measure of that vertex in relation to the other vertices?
Use [icon]https://www.geogebra.org/images/ggb/toolbar/mode_segment.png[/icon] to draw the segment from the [color=#cc0000][b]incenter[/b] [/color]to point D.[br]Use [icon]https://www.geogebra.org/images/ggb/toolbar/mode_segment.png[/icon] to draw the segment from the [color=#cc0000][b]incenter[/b][/color] to point E[br]Use [icon]https://www.geogebra.org/images/ggb/toolbar/mode_segment.png[/icon] to draw the segment from the [b][color=#cc0000]incenter[/color][/b] to point F.[br][br]3. These segments show the shortest distance from the [b][color=#cc0000]incenter[/color][/b] to each side of the triangle. Measure the angle between each segment and the triangle side it intersects. What do you notice?
Use [icon]/images/ggb/toolbar/mode_distance.png[/icon]to measure the length of each segment from the incenter.[br]Use [icon]/images/ggb/toolbar/mode_move.png[/icon] to drag a vertex of the triangle around. [br]4. What do you notice about the distance from the [color=#900000][b]incenter[/b][/color] to each side of the triangle?
[icon]/images/ggb/toolbar/mode_circle2.png[/icon]Construct a circle that is inside the triangle and touches each side of the triangle once. [br]5. Why do you think the name [b][color=#900000]incenter[/color][/b] was given to the point we are exploring in this activity?
6. For which of the following situations, would it make sense to find the [b][color=#900000]incenter[/color][/b]?