Circumscribed Circle (Circumcircle)
Definition
A circumscribed circle (or circumcircle) of a polygon is the circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter, and its radius is called the circumradius (denoted $R$).
Key Properties
For Triangles
- Every triangle has exactly one circumscribed circle.
- The circumcenter is the point where the perpendicular bisectors of all three sides meet.
- The circumcenter is equidistant from all three vertices.
- For an acute triangle: circumcenter is inside the triangle.
- For a right triangle: circumcenter is at the midpoint of the hypotenuse.
- For an obtuse triangle: circumcenter is outside the triangle.
Important Formulas
Circumradius Formula (using sides and area)
For a triangle with sides $a$, $b$, $c$ and area $A$:
$$R = \frac{abc}{4A}$$
Circumradius using the Law of Sines
$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R$$
For a Right Triangle
If the hypotenuse has length $c$:
$$R = \frac{c}{2}$$