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I have two concentric circles, one smaller and one larger. I need to geometrically construct a circular sector that will touch the outer and inner circles at its ends. The conditions are as follows:

  1. At the outer end, the angle between the sector and the tangent to the circle should be 30 degrees.
  2. At the inner circle, the sector should be perpendicular (i.e., it should point towards the center of both circles).

In terms of construction, this involves constructing a circle X that is tangential (touches line L at point P) and is also perpendicular to circle C (touches an imaginary line M that passes through the center of circle C, but we don't know this line; the imaginary line M is not parallel to line L).

I'm struggling with how to approach this problem. Any guidance or suggestions would be greatly appreciated.

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  • $\begingroup$ Something like this? i.stack.imgur.com/tGSP9.png $\endgroup$
    – Intelligenti pauca
    Jul 17 at 19:00
  • $\begingroup$ @Intelligentipauca Yes, that's exactly what i want to solve. I need to find the center of the circle (point E in this case). $\endgroup$
    – Břeťa
    Jul 17 at 20:21

1 Answer 1

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Let the given circles have common centre $A$ and radii $r=AD$, $R=AF$ (with $r<R$, see figure below).

We want to construct a third circle, with centre $E$ such that $DE\perp DA$ and radius $d=DE$, intersecting the outer circle at a point $F$, such that $\angle AFE=30°$. That is, we must find the value of $d$.

By the cosine law applied to triangle $AFE$ we get: $$ AE^2=EF^2+AF^2-2EF\cdot AF\cos30°, $$ that is: $$ d^2+r^2=d^2+R^2-dR\sqrt3. $$ From there one immediately gets $$ d={R^2-r^2\over \sqrt3\, R}. $$

enter image description here

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  • $\begingroup$ Thank you so much! It's a functional and very elegant solution. $\endgroup$
    – Břeťa
    Jul 18 at 7:37

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