### Constructions Involving Circles

```Constructions Involving
Circles
Section 7.4
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.
Constructing the Circumscribed
Circle of a Triangle
1. Construct a triangle
2. Construct the perpendicular bisector to any 2 sides
3. 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.
4. This point, the circumcenter, is equidistant from what?
Constructing the Inscribed Circle
of a Triangle
1. Construct a triangle.
2. Construct any two angle bisectors of the triangle.
3. The intersection of the angle bisectors is the incenter.
4. The incenter is equidistant from what?
Constructing the Centroid of a
Triangle
1. Construct a triangle.
2. 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.
3. The intersection of the two medians is the centroid.
4. What’s the importance of the centroid of a triangle?
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
Constructing the Orthocenter of a
Triangle
1. Construct a triangle.
2. 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.
3. The intersection of the altitudes is called the
orthocenter.
Constructing a Tangent to a Circle
at a Point on the Circle
1. Draw the radius to the given point, P.
2. Construct a perpendicular to the radius through point
P.

Use the construction “a perpendicular to a line through a
point on the line.”
Construct a Tangent to a Circle
from a Point Outside the Circle
1. Consider circle with center O and a point P outside the
circle.
2. Construct the midpoint M of OP.
3. Construct the circle with center M and radius MP and
MO.
4. Let Q be the point where the two circles intersect.
5. PQ will be tangent to circle O.
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.
Construct a Common Internal
Tangent to Two Circles
1. Consider circles O and P.
2. Draw OP, giving two points of intersection Q and R.
3. Using O as center, construct a circle whose radius is OQ +
PR.
4. Construct a tangent line from P to the new circle S using the
previous construction.
5. Draw OS, calling T the point where OS intersects the original
circle with center O.
6. Construct a line l, through T, parallel to SP. Line l is the
common internal tangent to the original two circles.
```