Inscribed and circumscribed circles of a triangle
Inscribed and Circumscribed Circles
Inscribed and Circumscribed Circles is a topic in Circles in the California Common Core State Standards. It is aligned to Standard G-C.3, which requires students to construct the inscribed and circumscribed circles of a triangle and prove properties of angles for a quadrilateral inscribed in a circle.
A triangle's incircle (centered at the incenter) is tangent to all three sides, and its circumcircle (centered at the circumcenter) passes through all three vertices.
Theory
Two circles are naturally associated with a triangle:
- The inscribed circle (incircle) is tangent to all three sides. Its center, the incenter, is where the angle bisectors meet, equidistant from the sides.
- The circumscribed circle (circumcircle) passes through all three vertices. Its center, the circumcenter, is where the perpendicular bisectors meet, equidistant from the vertices.
The centers:
How to locate each circle
- Incircle: construct the angle bisectors; their meeting is the incenter.
- Circumcircle: construct the perpendicular bisectors; their meeting is the circumcenter.
- Radius: incircle to a side; circumcircle to a vertex.
The incenter — where the angle bisectors meet; it is equidistant from the three sides.
The circumcenter — where the perpendicular bisectors meet; it is equidistant from the three vertices.
The inscribed circle is tangent to the sides; the circumscribed circle passes through the vertices.
At the midpoint of the hypotenuse; the circumradius is half the hypotenuse.
Common pitfalls
Frequently asked questions
What is an inscribed circle?
A circle inside a triangle tangent to all three sides; its center is the incenter.
What is a circumscribed circle?
A circle passing through all three vertices of a triangle; its center is the circumcenter.
Where is the incenter?
Where the three angle bisectors meet, equidistant from the sides.
Where is the circumcenter of a right triangle?
At the midpoint of the hypotenuse, so the circumradius is half the hypotenuse.