Inscribed Circle (Incircle) of a Triangle
📐 What is an Inscribed Circle?
An inscribed circle (also called an incircle) is the largest circle that fits completely inside a triangle, touching all three sides.
The inscribed circle is tangent to each side of the triangle, meaning it touches each side at exactly one point. The center of the inscribed circle is called the incenter.
🎯 Key Definitions
| Term | Definition |
|---|---|
| Incircle | The circle inscribed inside a triangle that is tangent to all three sides |
| Incenter | The center of the incircle; the point where all three angle bisectors meet |
| Inradius ($r$) | The radius of the inscribed circle |
| Angle Bisector | A line that divides an angle into two equal parts |
| Semi-perimeter ($s$) | Half the perimeter of the triangle: $s = \frac{a + b + c}{2}$ |
⭐ The Big Idea: Angle Bisectors
The incenter is found at the intersection of the triangle's three angle bisectors. This is a remarkable fact:
Theorem: The three angle bisectors of any triangle always meet at a single point (the incenter), and this point is equidistant from all three sides of the triangle.
Since the incenter is the same distance from all three sides, that distance becomes the inradius — the radius of the inscribed circle.
📏 Formulas for the Inscribed Circle
Formula 1: Inradius from Area and Semi-perimeter
$$r = \frac{A}{s}$$
where $A$ = area of the triangle and $s = \frac{a+b+c}{2}$ is the semi-perimeter
Formula 2: Area using Inradius
$$A = r \cdot s$$
The area of a triangle equals the inradius times the semi-perimeter