2 DefinitionsConcurrent: When three or more lines meet at a single pointCircumcenter of a Triangle: The point where the 3 perpendicular bisectors of a triangle intersect.Centroid of a Triangle: The point where the 3 medians of a triangle intersect.Incenter of a Triangle: The point where the 3 angle bisectors of a triangle intersect.Orthocenter of a Triangle: The point where the 3 altitudes of a triangle intersect.
3 Constructing the Circumscribed Circle of a Triangle Construct a triangleConstruct the perpendicular bisector to any 2 sidesThe intersection of the two perpendicular bisectors is the circumcenter of the triangle. If you place your compass tip at the intersection and extend it to a vertex on the triangle, you can construct a circumscribed circle.This point, the circumcenter, is equidistant from what?
4 Constructing the Inscribed Circle of a Triangle Construct a triangle.Construct any two angle bisectors of the triangle.The intersection of the angle bisectors is the incenter.The incenter is equidistant from what?
5 Constructing the Centroid of a Triangle Construct a triangle.Construct the midpoints of any two sides and draw the medians.Use the perpendicular bisector construction to find the two midpoints.A median connects a midpoint and the opposite vertex of a triangle.The intersection of the two medians is the centroid.What’s the importance of the centroid of a triangle?
6 Centroid Theorem In triangle ABC, if G is the centroid, then: CG = 2/3 CMC GMC = 1/3 CMCBG = 2/3 BMB GMB = 1/3 BMBAG = 2/3 AMA GMA = 1/3 AMA
7 Constructing the Orthocenter of a Triangle Construct a triangle.Construct two altitudes.Use the construction “perpendicular to a line through a given point not on the line”You might need to extend the sides of the triangle.The intersection of the altitudes is called the orthocenter.
8 Constructing a Tangent to a Circle at a Point on the Circle Draw the radius to the given point, P.Construct a perpendicular to the radius through point P.Use the construction “a perpendicular to a line through a point on the line.”
9 Construct a Tangent to a Circle from a Point Outside the Circle Consider circle with center O and a point P outside the circle.Construct the midpoint M of OP.Construct the circle with center M and radius MP and MO.Let Q be the point where the two circles intersect.PQ will be tangent to circle O.
10 DefinitionsCommon Internal Tangents to 2 Circles: If two circles do not intersect, then a tangent to the two circles that crosses the segment connecting the two centers of the circles is called a common internal tangent.Common External Tangent to 2 Circles: If two circles do not intersect, then a tangent to the two circles that does not cross the segment connecting the two centers of the circles is called a common external tangent.
11 Construct a Common Internal Tangent to Two Circles Consider circles O and P.Draw OP, giving two points of intersection Q and R.Using O as center, construct a circle whose radius is OQ + PR.Construct a tangent line from P to the new circle S using the previous construction.Draw OS, calling T the point where OS intersects the original circle with center O.Construct a line l, through T, parallel to SP. Line l is the common internal tangent to the original two circles.