Circumscribed circle

Circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter, and its radius is known as the circumradius. Circumscribed circles are significant in geometry, particularly in the study of polygons and triangles, where they are used to explore relationships between angles, sides, and other geometric properties.

Definition

A circumscribed circle of a polygon is a circle that contains all the vertices (corner points) of the polygon on its circumference. The polygon is then said to be inscribed in the circle. If such a circle exists, the polygon is called cyclic or concyclic. Not all polygons have a circumscribed circle; for a polygon to have a circumscribed circle, it must satisfy certain properties.

Properties

• For a triangle, a circumscribed circle always exists. The circumcenter of a triangle can be found as the intersection point of the perpendicular bisectors of the sides of the triangle.
• In the case of a quadrilateral, a circumscribed circle exists if and only if the sum of the measures of the opposite angles is 180 degrees. Such quadrilaterals are called cyclic quadrilaterals.
• For polygons with more than four sides, determining the existence of a circumscribed circle involves more complex conditions related to the polygon's angles and sides.

Construction

The construction of a circumscribed circle varies based on the type of polygon. For triangles, the circumcenter can be constructed by drawing the perpendicular bisectors of at least two sides of the triangle. The point where these bisectors intersect is the circumcenter, and the circle drawn with this point as the center and extending to any of the triangle's vertices is the circumscribed circle.

Applications

Circumscribed circles have applications in various fields of mathematics and science, including:

• In geometry, for solving problems related to angles, distances, and areas.
• In trigonometry, circumscribed circles are used to derive various trigonometric identities.
• In engineering and architecture, understanding the properties of circumscribed circles aids in the design of structures and components.

See also

• Inscribed circle
• Circumcenter
• Circumradius
• Cyclic quadrilateral