Inscribed and Circumscribed Circles of a Triangle
Big Idea: Every triangle has two special circles associated with it — one that fits perfectly inside the triangle (the inscribed circle) and one that passes through all three vertices (the circumscribed circle). Understanding these circles reveals deep connections between a triangle's sides, angles, and area.
📐 The Inscribed Circle (Incircle)
Definition
The inscribed circle (or incircle) of a triangle is the largest circle that fits entirely inside the triangle. It is tangent (just touches) all three sides of the triangle.
Key Properties
- The center of the incircle is called the incenter, denoted by $I$.
- The incenter is the point where all three angle bisectors of the triangle meet.
- The inradius $r$ is the perpendicular distance from the incenter to any side.
- The incenter is always inside the triangle, no matter what type of triangle it is.
Formula for the Inradius
If the triangle has area $A$ and semi-perimeter $s = \frac{a + b + c}{2}$, then:
$$r = \frac{A}{s}$$
🔵 The Circumscribed Circle (Circumcircle)
Definition
The circumscribed circle (or circumcircle) of a triangle is the circle that passes through all three vertices of the triangle.
Key Properties
- The center of the circumcircle is called the circumcenter, denoted by $O$.
- The circumcenter is the point where all three perpendicular bisectors of the sides meet.
- The circumradius $R$ is the distance from the circumcenter to any vertex.
- The circumcenter can be inside (acute △), on the hypotenuse (right △), or outside (obtuse △).
Formula for the Circumradius
If the triangle has sides $a$, $b$, $c$ and area $A$, then:
$$R = \frac{abc}{4A}$$