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Math Help - Finding tangential line between 2 circles.

  1. #1
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    Finding tangential line between 2 circles.

    I am trying to find a line that crosses in between two circles and is tangential to both. To help visualize, imagine a conveyor belt passing between two rollers. I have attached an image to explain it properly.

    I know the radius of each circle and the distance between them. The problem is that each tangential point is determined by the other, so it seems it would be solved using a simultaneous equation, but I don't know the mathematics to work this out.

    Any help would be appreciated.
    Attached Thumbnails Attached Thumbnails Finding tangential line between 2 circles.-circleproblem.jpg  
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  2. #2
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    Re: Finding tangential line between 2 circles.

    Quote Originally Posted by TrevDA View Post
    I am trying to find a line that crosses in between two circles and is tangential to both. To help visualize, imagine a conveyor belt passing between two rollers. I have attached an image to explain it properly.

    I know the radius of each circle and the distance between them. The problem is that each tangential point is determined by the other, so it seems it would be solved using a simultaneous equation, but I don't know the mathematics to work this out.

    Any help would be appreciated.
    1. I've attached a sketch how to find the common inner tangents of 2 circles.

    2. Use an auxiliar circle whose radius is the sum of the two radii. Draw the tangent from the center of the left circle to this aux. circle. Since the tangent and the radius form a right angle at the tangent point you have to use Thales' circle (dotted line).

    3. Translate this tangent until it touches both circles simultaneously.
    Attached Thumbnails Attached Thumbnails Finding tangential line between 2 circles.-zweikrs_innentang.png  
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  3. #3
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    Re: Finding tangential line between 2 circles.

    Wow! That's a fantastic solution - exactly what I need. Thanks very much for your help.
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  4. #4
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    Re: Finding tangential line between 2 circles.

    Hi TrevDA,
    Here is an example of a math solution

    given d= 10 cm distance between centers
    r1 =4 cm r2 = 1 cm
    Find length of common inner tangent

    Draw line of centers and the two radii from each center to points of tangency.The right triangles formed are similar (AAA)

    4/1 =d1/10 -d1 d1=8 segment of line of centers
    4^2+t1^2 =8^2 t1 = rad 48 segment of tangent
    4/1=rad 48/t2 t2 = rad 48/4 = 4rad3/ 4 =rad 3 other segment of tangent
    t1 +t2 = 4rad3 +rad3 =5rad3
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