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6.9 – Circles and Triangles
Key Terms
 Circumscribed – To fit an object tightly around another.
 Inscribed – To fit one object inside another.
Notes
Circumscribed – See the red figures in the images below 
Inscribed – See the blue figures in the images below 
 To draw a figure around another, touching it at points but not cutting through it.
 The figure on the outside is circumscribed about the figure on the inside.
 The figure on the inside is inscribed.
 Exactly one circle can be circumscribed about a figure (see circles in red, below).
 When a circle is circumscribed about a polygon, the polygon has a circumcenter.
 Each vertex of the figure must touch the circle.
 Circles Circumscribed About a Triangle
 Circumscribed figures have circumcenters.
 The circle is around the entire triangle.
 Exactly one circle can be circumscribed about a triangle.
 Each vertex of the triangle touches the circle.
 The shortest distance from the center of the circumscribed circle to the vertices of the inscribed triangle is the circle’s radius.
 The center of the circumscribed circle about a triangle is equidistant to the vertices of the inscribed triangle.
 Many figures can be circumscribed about a given circle (see polygons in red, below).
 These figures touch the circle at a point of tangency.
 Triangles Circumscribed About a Circle
 The triangle is around the entire circle.
 The circle is tangent to each side of the triangle.
 Many triangles can be circumscribed about a given circle.

 To draw a figure inside of another, touching it at points, but not cutting through it.
 The figure on the inside is inscribed inside the other figure.
 The figure on the outside is circumscribed about it.
 Inscribed figures have incenters, the center of the circle that is circumscribed about it (see figures in blue, below).
 Ex. A triangle is inscribed in another figure if each vertex of the triangle touches that figure.
 Ex. A square is inscribed in another figure if each corner of the square touches the figure.
 Circles can be inscribed inside other polygons (see figures in blue, below).
 These polygons touch the circle at a point of tangency.

 Inscribed and Circumscribed
 If figure X is inscribed in figure Y, figure Y is circumscribed about figure X.
 If polygon X is inscribed in polygon Y, polygon Y is circumscribed about polygon X.

Circumcenter – When the Circle is Circumscribed (Outside) 
Incenter – When the Circle is Inscribed (Inside) 
 The center of the only circle that can be circumscribed about a figure.
 Mostly commonly, this figure is a triangle.
 Equidistant from all vertices of the figure.

 The center of the only circle that can be inscribed in a figure.
 The incenter of a figure is the point equidistant from each side of the figure.
 Ex. The incenter of a triangle is the center of the only circle that can be inscribed in a triangle.

Triangle Inscribed in a Circle 
Circle Inscribed in a Triangle 
 Entire triangle is inside the circle.
 Each vertex of the triangle touches the circle.
 Many triangles can be inscribed in a given circle.
 The center of the only circle that can be circumscribed about the triangle is called the circumcenter.

 Entire circle is inside the triangle.
 The circle is tangent to each side of the triangle.
 Exactly ONE circle can be inscribed in a given triangle.
 The shortest distance from the center of the inscribed circle to the triangle’s sides is the circle’s radius.
 In order to inscribe a circle in a triangle, the circle’s center must be placed at the incenter of the triangle.

 Properties of the Circumcenter of a Triangle
 It is at the intersection of the perpendicular bisectors of the triangle’s sides (see them in red, blue, and green below).
 It is equidistant from each vertex of the triangle.
 … because that distance is the radius of the circle.
 It is the center of the only circle that can be circumscribed about it.
 Location of the Circumcenter as it relates to an inscribed triangle:

 Properties of the Incenter of a Triangle
 The incenter is the point where all of the bisectors of the angles of the triangle meet.
 The incenter of a triangle is always inside it.
 The incenter is equidistant from each side of the triangle.
 How to Find the Incenter of a Triangle
 Draw the angle bisector for each vertex.
 Find the point where the three bisectors intersect.
 That point is the incenter!

 Equilateral Triangles
 Their incenters and circumcenters are the same point

Important!
Practice (Apex Study 6.9)
 Try practice problems on Pgs 8, 28
 Mandatory: write and answer problems on Pg 29
 1 Quiz
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