1, circumscribed circle: The circle that intersects all vertices of a polygon is called the circumscribed circle of the polygon, which is usually used for convex polygons, such as triangles. If a circle just passes through three vertices, it is called the circumscribed circle of a triangle, and the circle just surrounds the triangle.

2. Inscribed circle: Mathematically, if each side of a polygon on a two-dimensional plane can be tangent to a circle inside it, then this circle is the inscribed circle of this polygon, and then this polygon is called a circle circumscribed polygon. It is also the largest circle in a polygon. The center of the inscribed circle is called the center of the polygon.

3, inscribed circle: usually for another circle, if a circle is in another big circle, the two circles have only one common point, this circle is called inscribed circle of the big circle.

4. circumscribed circle: circumscribed circle refers to another circle. If two circles have only one common point and the distance between the centers is equal to the sum of the radii of the two circles, the two circles are tangent to each other. When two circles are circumscribed, there are three common tangents.

Second, the painting method.

1, circumscribed circle: that is, the median vertical line of three sides of a triangle (two lines can also intersect to determine a point).

Take a line segment as an example, which can be regarded as an edge of a triangle. Take two endpoints as the center, make a circle with an appropriate length (equal) as the radius (only draw an arc intersecting the line segment), then make a circle with the same length with two intersections as the center (ensure the intersection of the two circles), and make a straight line through the two intersections of the last two circles. This straight line is vertical, bisecting the line segment, that is, the midline of the line segment.

2. Inscribed circle: In a triangle, the intersection of bisectors of three angles is the center of the inscribed circle, and the vertical segments from the center to each side of the triangle are equal. A regular polygon must have an inscribed circle, and the center of the inscribed circle coincides with the center of the circumscribed circle, both of which are at the center of the regular polygon.

3. inscribed circle: the circle tangent to all three sides of a triangle is called the inscribed circle of the triangle, the center of the circle is called the heart of the triangle, and the triangle is called the circumscribed triangle of the circle. The center of a triangle is the intersection of three bisectors of the triangle.

4. circumscribed circle: the point connecting the center of the circle and the point outside the circle intersects with the circle at a point, and the circle is drawn with the point outside the circle as the radius and the point outside the circle as the center.

Third, restrictions.

1, circumscribed circle, triangle has circumscribed circle, other figures do not necessarily have circumscribed circle. The center of the circumscribed circle of a triangle is the intersection of perpendicular bisector of any two sides. The center of the circumscribed circle of a triangle is called the outer center.

2. The circle that the circumscribed circle intersects with each vertex of the polygon is called the circumscribed circle of the polygon. Geometry is in a circle, and its vertex is in this circle.

3. inscribed circle: a polygon has at most one inscribed circle, which means that for a polygon, its inscribed circle, if it exists, is unique. Not all polygons have inscribed circles. Triangles and regular polygons must have inscribed circles. A quadrilateral with inscribed circle is called a circumscribed quadrilateral.

4, inscribed circle, triangle must have inscribed circle, other graphics do not necessarily have inscribed circle, inscribed circle center is located inside the triangle.