Let P be a point on the curve, if astraight linePassing through P and another adjacent point Q on the curve, PQ is called the secant of the curve.When Q tends to P, the limit position of the secant PQ is called thetangent, P is calledTangency point, the line and curveTangencyAt point P.If each side of a polygon (or polyhedron) is tangent to a closed curve (or surface) within it, it is called a polygon (or polyhedron)CircumscribeOn this curve (or surface).
Two circles have only one common point, which is called the tangency of two circles, and the common point is calledTangency point, there are two kinds of tangency between two circles[1]：
(1) BicircleCircumscribe, as shown in Figure 1 (a);
The straight line connecting the centers of two circles is calledCentripetal lineWhen two circles are tangent, the tangent point is on the centerline.
When two circles are circumscribed, the distance between the center of the circle O ₁ O ₂=R+r, (let the radius of the large circle be R, and the radius of the small circle be r).
When two circles are inscribed, the center distance O ₁ O ₂=R-r[1]。
The connecting center line or its extension line of two tangent circles must pass through the tangent point.
As shown in figure (a), ⊙ O ₁, and ⊙ O ₂ are tangent to point T, then the connecting line O ₁ O ₂ must pass through point T.
As shown in Figure (b), ⊙ O ₁, and ⊙ O ₂ are tangent to point T, then the extension line of connecting line O ₁ O ₂ must pass through point T.
Circumtangent of circle and polygon
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Fig. 2 Tangency between circle and polygon
Fig. 3 Tangency between circle and polygon
Circumtangent of circlepolygon: If a circle is a polygonInscribed circleAll sides of the polygon are tangent to a circle, which is called the circumscribed polygon of the circle, and the circle is called the inscribed circle of the polygon.
As shown in the figure, the pentagon ABCDE is the circumscribed pentagon of circle O.
For example, the quadrilateral ABCD in the figure is the circumscribed quadrilateral of ⊙ O, while ⊙ O is the inscribed circle of the quadrilateral ABCD.
Two ball circumscission
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There are two balls, whose centers are
, the radii are
, center distance
；So there are:
one
The two balls only have one or two common points, which are on the connecting heart.If you cross this common point and make a plane perpendicular to the centerline, then two balls will be cut on both sides of this plane, and they are tangent to this plane. Such two balls are calledTwo ball circumscission。
two
The two balls have only one common point, which is on the extension line of the connecting line. If a plane is perpendicular to the connecting line after passing this common point, then the two balls are on the same side of A1: plane, and they are both connected with this plane. Such two balls are calledTwo balls inscribed。
three
When the two balls have no common points, all the points on one ball are inside the other ball, and all the dots on the other ball are outside the ball. Such two balls are calledTwo balls inside out。
four
There is no common point between the two balls.All points on any ball are outside the other ball. Such two balls are calledTwo balls apart。
five
Hour;Two balls have countless common points, which form a circle. Such two balls are calledintersect。
Circumscribed polyhedron of sphere
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The inscribed sphere of a polyhedron is a sphere that meets certain conditions. If a sphere is tangent to all faces of a simple polyhedron or its extension, and the sphere is inside the polyhedron, then the sphere is said to be theInscribed ballThe polyhedron is called the sphericalcircumscribed polyhedron。The inscribed sphere of a regular polyhedron exists. The sum of the distances from any point in the regular polyhedron to each face is a constant 3FV/S. Here, F is the number of faces of the polyhedron, S is the surface area, and V is the volume. Therefore, the radius of the inscribed sphere of a regular polyhedron is 3V/S.
Examples of other circumscribed situations
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Circumscribed prism of a column
Circumscribed prism of a column(circumscribed prism of a cylinder) is a prism related to a known column.The prism that meets the following conditions is called the circumscribed prism of the prism (the prism is called the inscribed prism of the prism):
1. The two bottoms of the prism are the circumscribed polygons of the corresponding bottom of the column (that is, the edge of the bottom of the prism is tangent to the boundary line of the bottom of the column);
2. The side edges of the prism are parallel and equal to the generatrix of the column.
Circumscribed pyramid of a cone
Circumscribed pyramid of a cone(circumscribed pyramid of a cone) is a pyramid related to a known cone, that is, a pyramid circumscribed with the cone.The pyramid meeting the following conditions is called the circumscribed pyramid of the cone (body) (the cone is called the inscribed cone of the pyramid):
1. The bottom polygon of the pyramid is circumscribed to the bottom curve of the pyramid;
