[size=150]Kolton’s father installs sprinkling systems for farmers. Sometimes Kolton’s father has to install sprinkler systems on triangular-shaped pieces of land. He wants to be able to locate the “pivot point” (INCENTER) in the triangular field so the circle being watered will touch each of the three fences that form the boundaries of the field.[br][br][img]https://mathbitsnotebook.com/Geometry/Constructions/incenter1.jpg[/img][br][br][/size][size=150]Learn more about the [b]incenter[/b] [url=https://mathbitsnotebook.com/Geometry/Constructions/CCIncenter.html]here[/url].[/size]

[size=150]Go to [url=https://www.nctm.org/Classroom-Resources/Illuminations/Interactives/WSP-Incenter-Incircle/][b]this[/b][/url] website and explore how to find the [b]INCENTER[/b] and build the [b]INCIRCLE[/b].[/size]

[size=150]I constructed the angle bisectors of angle ACB and angle ABC. They intersected at point P, the [b]INCENTER[/b]. So I wouldn’t get confused by so many lines in my diagram, I erased the rays that formed the angle bisectors past their point of intersection. I then drew ray AH through point P, the point of intersection of the two angle bisectors. My question is, “[b]Does this ray bisect angle CAB[/b]?”[br][br][img]https://i.imgur.com/soi38ur.png[/img][br][br][/size][size=150]In order to prove that segment AH bisects [math]\angle[/math]FAG, we need to prove [math]\angle[/math]FAP[math]\cong[/math][math]\angle[/math]GAP by demonstrating that triangles [math]\bigtriangleup[/math]FAP and [math]\bigtriangleup[/math]GAP are congruent.[/size]