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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.

GeoB 6.9 Circle Circumscribed About Figures

  • 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.

GeoB 6.9 Figures Circumscribed About a Circle

  • 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.

GeoB 6.9 Circle Circumscribed About Figures

  • Circles can be inscribed inside other polygons (see figures in blue, below).
    • These polygons touch the circle at a point of tangency.

GeoB 6.9 Figures Circumscribed About a Circle

  • 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.

GeoB 6.9 Chart

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.

GeoB 6.9 Circumcenter2

  • 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.

GeoB 6.9 Incenter

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.

GeoB 6.9 Triangle Inscribed in a Circle

  • 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.

GeoB 6.9 Circle Inscribed in a 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:

GeoB 6.9 Circumcenter

GeoB 6.9 Acute GeoB 6.9 Obtuse GeoB 6.9 Right

  • 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!

GeoB 6.9 Incenter Angle Bisector

 

  • Equilateral Triangles
    • Their incenters and circumcenters are the same point

GeoB 6.9 Equilateral


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