Inscribed and circumscribed circles of a triangle
Inscribed and Circumscribed Circles
Inscribed and Circumscribed Circles is a topic in Circles in the Texas Essential Knowledge and Skills (Geometry, §111.41). It is aligned to Standard G.12(A), which requires students to apply theorems about the inscribed and circumscribed circles of a triangle.
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.