Finding center of a circle given limited infomation

[texasaggie0004]

Good morning,

I'm usually able to mechanically solve these types of problems, and maybe I've over complicated myself here and am missing the simple solution.

Please refer to the attached image. I have a circle (black) with a known radius and center. I have another known point off the circle. Given the limits of the red lines (known by drawing right angles off the center of the known circle and the location of the known point - is there enough information to determine the center and radius of a circle (green) that is tangent to both the known point, the known circle AND whose own center lies on the red line indicated in the illustration?

Thanks!

 

[texasaggie0004]

Ah - found the solution - I can reflect the known circle across the uppermost red line and do the mechanical tasks for find a circle tangent to the now, 3 known objects.

 

[lev888]

Ah - found the solution - I can reflect the known circle across the uppermost red line and do the mechanical tasks for find a circle tangent to the now, 3 known objects.

I don't see how it helps. You can construct multiple green circles tangent to the black circle and through the known point.

 

[Dr.Peterson]

Good morning,

I'm usually able to mechanically solve these types of problems, and maybe I've over complicated myself here and am missing the simple solution.

Please refer to the attached image. I have a circle (black) with a known radius and center. I have another known point off the circle. Given the limits of the red lines (known by drawing right angles off the center of the known circle and the location of the known point - is there enough information to determine the center and radius of a circle (green) that is tangent to both the known point, the known circle AND whose own center lies on the red line indicated in the illustration?

Thanks!

I'm not sure I understand some of your terms (e.g. "limits of the red lines").

As I understand it, you are given a circle (center A, known radius AB in my picture below), and a point C outside the circle, and a given line through C (red in my picture). You want to construct a circle (green) with its center F on the given line, such that it passes through point C, and is tangent to the given circle:

​

The drawing illustrates one way to construct F; I can't tell what your method is in order to see if I agree with it. I'm sure there are many ways to do it.

 

[lev888]

I don't see how it helps. You can construct multiple green circles tangent to the black circle and through the known point.

I take it back - played with CAD, there is a unique solution.

 

[texasaggie0004]

I'm not sure I understand some of your terms (e.g. "limits of the red lines").

As I understand it, you are given a circle (center A, known radius AB in my picture below), and a point C outside the circle, and a given line through C (red in my picture). You want to construct a circle (green) with its center F on the given line, such that it passes through point C, and is tangent to the given circle:

View attachment 38019​

The drawing illustrates one way to construct F; I can't tell what your method is in order to see if I agree with it. I'm sure there are many ways to do it.

I don't think that's a solution because we don't know the angle FAB in your illustration.

 

[texasaggie0004]

I take it back - played with CAD, there is a unique solution.

I ultimately realized, also in CAD, that reflecting the known circle across the upper red line (the one that would go through my known outer point AND through the unknown center of the unknown circle) that I'd get a situation where I could do the Circle from 3 tangents to solve the problem.
Which could be done by hand (one of Apollonius' problems).

It is for an architectural situation (trying to resolve some curves with a true radius and not just a randomly guessed curve). I strongly suspected there was only 1 solution that was on "that side" of the known circle and point.

Thanks for the confirmation!

 

[texasaggie0004]

I'm not sure I understand some of your terms (e.g. "limits of the red lines").

As I understand it, you are given a circle (center A, known radius AB in my picture below), and a point C outside the circle, and a given line through C (red in my picture). You want to construct a circle (green) with its center F on the given line, such that it passes through point C, and is tangent to the given circle:

View attachment 38019​

The drawing illustrates one way to construct F; I can't tell what your method is in order to see if I agree with it. I'm sure there are many ways to do it.

I meant to say, I wanted the center of the unknown circle to be on the same line as the known point - in such a way that those other red lines related by right angles to the center of the known circle. I know the term "limits" means a lot different in math - but for this practical exercise the red lines were limiting the options for where the unknown circle could be drawn.

Thanks for the input!

 

[blamocur]

I don't think that's a solution because we don't know the angle FAB in your illustration

Why do you need that angle?
Hint: DF = AF ==> EF is orthogonal to AD.

 

[Dr.Peterson]

I don't think that's a solution because we don't know the angle FAB in your illustration.

You aren't given it, but I constructed it using what was given! Why would that not be a solution??

Of course, I haven't stated what I did, but just showed a sequence of points I constructed in alphabetical order; so you are probably just making a wrong guess. If you hadn't said you'd already found a solution, I would have said more.

I ultimately realized, also in CAD, that reflecting the known circle across the upper red line (the one that would go through my known outer point AND through the unknown center of the unknown circle) that I'd get a situation where I could do the Circle from 3 tangents to solve the problem.
Which could be done by hand (one of Apollonius' problems).

It is for an architectural situation (trying to resolve some curves with a true radius and not just a randomly guessed curve). I strongly suspected there was only 1 solution that was on "that side" of the known circle and point.

You haven't made it clear what sort of construction is allowed when you "mechanically solve" a problem; I assume it is not much different from the compass and straightedge construction I interpreted it as. Evidently, though, you are reducing the problem to a known problem (Apollonius), which is a nice approach.

Reference to Apollonius also explains why you were willing to say the circle is "tangent" to a point! (Of course, that doesn't tell me what specific solution to the problem you have in mind, as there are many.)

I meant to say, I wanted the center of the unknown circle to be on the same line as the known point - in such a way that those other red lines related by right angles to the center of the known circle. I know the term "limits" means a lot different in math - but for this practical exercise the red lines were limiting the options for where the unknown circle could be drawn.

Thanks for the input!

It is still not at all clear what those "other red lines" have to do with the problem. I'll assume, until told otherwise, that they don't invalidate what I did.

 

[texasaggie0004]

You aren't given it, but I constructed it using what was given! Why would that not be a solution??

Of course, I haven't stated what I did, but just showed a sequence of points I constructed in alphabetical order; so you are probably just making a wrong guess. If you hadn't said you'd already found a solution, I would have said more.


You haven't made it clear what sort of construction is allowed when you "mechanically solve" a problem; I assume it is not much different from the compass and straightedge construction I interpreted it as. Evidently, though, you are reducing the problem to a known problem (Apollonius), which is a nice approach.

Reference to Apollonius also explains why you were willing to say the circle is "tangent" to a point! (Of course, that doesn't tell me what specific solution to the problem you have in mind, as there are many.)


It is still not at all clear what those "other red lines" have to do with the problem. I'll assume, until told otherwise, that they don't invalidate what I did.

I'm not engaging snark. Thanks for the previous input. Have a good day.