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I have been practicing my straightedge and compass constructions over the last few days and I'am trying to reproduce the solutions to the ten Apollonius problems (constructing circles which are tangent to three given objects which can be some combination of points, lines, and circles).

I'm doing fine for everything except LLC, the two lines and one circle case. Every site online that I can find says there are 8 solutions in the general case but I can only find 4. No site online seems to do the full construction of all 8 solutions, and after trying this one problem for over a day I'm starting to think it may be in error.

So I thought I would turn to stackexchange and ask if anyone knows a resource that constructs all 8 circles solving the Line-Line-Circle Apollonius problem?

Or if anyone knows whether Type 8: One circle, two lines and Apollonius problems all just have a typo?

Even just a picture with 8 solutions would help me out immensely.

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You'll have eight solutions only for some positions of given lines and circle, see picture below for an example.

enter image description here

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  • $\begingroup$ Thank you very much. I wasn't using a large enough circle, I suppose. Mine was crossing only one line at most. Very nice picture by the way. $\endgroup$ – Allen O'Hara Apr 8 '17 at 17:44
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Here is one with smaller circles:LLC

In the same way that you cannot construct two tangents to a circle from an interior point because the coefficients in the lines are not real, there will not be eight circles with real centres and radii for all initial LLC unless the circle intersects each of the quadrants formed by the lines.

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