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kelaklub

common tangent

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This is driving me nuts. I just figured out how to state the problem, now I was wondering if someone could point me in the rigth direction to solve it. First a definition... A common tangent to two circles is called a direct common tangent if both the circles lie on the same side of it, and it is called a transverse (or indirect) common tangent if the circles lie on opposite sides of it. I have 2 circles, a little one and a big one. Highly detailed sketch... o O I know that if r1, and r2 are the radii of two circles and d is the distance between their centers, then the length of a direct common tangent is sqrt(d^2 -(r1 - r2)^2) How do I get the two points that lie on the direct common tangent of this circle? Thank you.

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Its simple.

Lets define some stuff:
z is the length of the tangent line segment.
R1 is the center of one circle (radius == r1).
R2 is the center of the other circle (radius == r2).
T1 is the point where the tangent line is tangent to circle at R1.
T2 is the point where the tangent line is tangent to circle at R2.


You have your right triangle with sides z, d, and (r1-r2). The angle between the segments R1R2 and R1T1 (and R2T2) is sin-1(z/d).

edit: typos

[edited by - JohnBolton on September 2, 2003 9:03:14 PM]

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Guest Anonymous Poster
Big circle is centered on C0 with radius R0, little circle at C1 with R1. The direct tangent intersects circles at T0 and T1 respectively. If you draw the line from C0 through C1 extending indefinitely, and the line from T0 through T1 extending indefinitely, if R0 != R1, then the lines will intersect, forming 2 similar right triangles (the third leg of each being the radius of each circle drawn from Cx to Tx). Call this intersection point C2. Our goal is to find theta -- the angle of the triangle at C0.

Call D0 the distance from C0 to C2 and D1 is from C1 to C2, and the distance from C0 to C1 X (I''m running out of letters here )

we know the following:
cos(theta) = R0/D0
D0 = D1 + x
D0/D1 = R0/R1

X, R0, and R1 are knowns, so to solve for theta you must solve for D0. Luckily, we have 2 equations and 2 unknowns which is sufficient to solve the problem.

adam

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