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.