As a furtherexploration,try the followingwith paper:
- Cut out any trianglefrom a sheetof paper.
- Fold the the triangleover one side so that the side is foldedin half.
- Repeatfor the othertwo sides.
- Whatdo you notice?
Noticethat the foldswill crossat the circumcenter, unlessthe triangleis obtuse.In whichcasethe fold lines
will meetoutsidethe triangleif they continued.
LessonSummary
In this lessonwe:
- Definedthe perpendicularbisectorof a line segment.
- Statedand provedthe PerpendicularBisectorTheorem.
- Solvedproblemsusingthe PerpendicularBisectorTheorem.
Pointsto Consider
If we thinkaboutthreenon-collinearpointsin a plane,we can imaginea trianglethat has eachpointas a
vertex.Locatingthe circumcenter, we can drawa circlethat all threeverticeswill be on. Whatdoesthis tell
us aboutany threenon-collinearpointsin a plane?
Thereis a uniquecirclefor any threenon-collinearpointsin the sameplane.
Findinga circlethroughany threepointswill also work in coordinategeometry. You can use the circumcenter
to find the equationof a circlethroughany threepoints.In calculusthis methodis used(togetherwith some
toolsthat you haveprobablynot learnedyet) to preciselydescribethe curvatureof any curve.
LessonExercises
Constructthe circumcenterof and the circumscribedcirclefor eachof the followingtrianglesusing
a straightedge,compass,and Geometer'sSketchpad.