Definition of Common Tangent Lines
A common tangent line is a line that touches two circles simultaneously. There are two types of common tangent lines:
- External Common Tangent Line: A line that touches both circles from the same side
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A common tangent line is a line that touches two circles simultaneously. There are two types of common tangent lines:
An external common tangent line touches both circles without crossing the segment between their centers.
For two circles with:
Length of external common tangent line:
This positive-length segment exists when . At equality, the circles are internally tangent and share only the tangent line through the contact point.
Two circles are centered at with radius and with radius . Find the length of the external common tangent line!
Solution:
The length of the external common tangent line is .
An internal common tangent line is a line that touches both circles from opposite sides and intersects the line connecting the two circle centers.
Length of internal common tangent line:
Condition: A positive-length internal tangent segment exists when . At equality, the circles are externally tangent: the two points of tangency coincide, the segment length is zero, and only one internal common tangent remains.
Two circles are centered at with radius and with radius . Find the length of the internal common tangent line!
Solution:
First, check if internal common tangent lines exist:
Since the condition is satisfied, then:
The length of the internal common tangent line is .
When two circles have equal radii (), there are special properties:
For :
Determine the length of external and internal common tangent lines for the following circles:
The first circle is centered at with radius , the second circle is centered at with radius .
Solution:
The first circle is centered at with radius , the second circle is centered at with radius .
Solution:
Two circles are centered at with radius and with radius . Determine:
The first circle has center with radius , the second circle has center with radius . Calculate the length of both types of common tangent lines!
Two identical circles each have radius . If the length of the internal common tangent line is , determine the distance between the two circle centers!
Circle is centered at with radius , and circle is centered at with radius . If the length of the external common tangent line is , determine the value of !
Determine the conditions for two circles to have:
Calculating common tangent line lengths
Circles with different centers
Finding center distance from internal tangent length
Given: ,
Finding the value of
Given: , , ,
Conditions for number of common tangent lines
Let be the distance between centers. The complete classification for two distinct, noncoincident circles is:
| Center-distance condition | Common tangents | Circle position |
|---|---|---|
Published: . Updated: .
| externally disjoint |
| externally tangent |
| intersecting at two points |
| internally tangent |
| one circle strictly inside the other |
Therefore, the requested conditions are the first three rows. Coincident circles, with and , are a separate case with infinitely many common tangents.