# 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