1. Constructions Involving Circles
Section 7.4

2. Definitions
Concurrent: When three or more lines meet at a single point
Circumcenter 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 triangle
Construct the perpendicular bisector to any 2 sides
The 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 CMC
BG = 2/3 BMB GMB = 1/3 BMB
AG = 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. Definitions
Common 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.